Day By Day Notes for MATH 201

Fall 2006

 

Day 1

Activity: Go over syllabus.  Take roll.  Overview examples: Gilbert trial, election polls, spam filters

Goals:     Review course objectives: collect data, summarize information, make inferences.

Reading: To The Student, pages xxxi-xxxiv.

Day 2

Activity: Discussion of variables and graphs.  From a list of numbers, communicate the important information to the person next to you.  (Work in pairs or groups.)  For your list of numbers, make a frequency table and a histogram.


Useful commands for the calculator:
STAT EDIT (use one of the lists to enter data, L1 for example; the other L's can be used too.)
2nd STATPLOT 1 On (Use this screen to designate the plot settings.  You can have up to three plots on the screen at once.  For now we will only use one at a time.)
ZOOM 9 This command centers the window around your data.

 In your description to your neighbor, keep in mind these terms:  symmetry, skew, center, spread, mode, outlier.  Also make sure that you try different window settings for your histogram.

Goals:     Begin graphical summaries (describing data with pictures).  Be able to use the calculator to make a histogram.

Skills:

…                        Identify types of variables.  To choose the proper graphical displays, it is important to be able to differentiate between Categorical (or Qualitative) and Quantitative (or Numerical) variables.

…                        Be familiar with types of graphs.  To graph categorical variables we use bar graphs or pie graphs.  To graph numerical variables, we use histograms, stem plots, or QUANTILE (a TI-83 program we will explore on Day 3).  In practice, most of our variables will be numerical but it is still important to choose the right display.

…                        Summarize data into a frequency table.  The easiest way to make a frequency table is to first use the TI-83 to make a histogram and then to TRACE over the boxes and record the classes and counts.  You can control the size and number of the classes with Xscl and Xmin in the WINDOW menu.  The decision as to how many classes to create is arbitrary; there isn't a "right" answer, or rather all choices of Xscl and Xmin are "right" answers.  One popular suggestion is try the square root of the number of data values.  For example, if there are 25 data points, use 5 intervals.  If there are 50 data points, try 7 intervals.  This is a rough rule; you should experiment with it.  The TI-83 has a rule for doing this; I do not know what their rule is.  You should develop your intuitions by changing the interval width Xscl and starting point Xmin and see what happens to the display.

…                        Know how to create and interpret graphs for categorical variables.  The two main graphs for categorical variables are pie graphs and bar charts.  Pie graphs are difficult to make by hand, but popular on computer programs like Excel.  Bar charts are also common on spreadsheets.  Data represented by pie graphs and bar charts usually are expressed as percents of the whole; thus they add to 100%.  The ordering of categories is arbitrary; therefore concepts such as skew and center make no sense.

Reading: Section 1.1. (Skip Time Plots and Time Series.)

Day 3

Activity: Use the "Arizona Temps" dataset to practice creating the histograms, stem plots, and quantile plots for several lists.  Compare and interpret the graphs.  Identify shape, center, and spread.
    

              QUANTILE is a program I wrote that plots the sorted data in a list and "stacks" them up.  This is also known as a quantile plot.  Basically we are graphing the data value versus its rank, or percentile, in the dataset.  The syntax is PRGM EXEC QUANTILE ENTER.

Answer these questions:

1)  Do any of the lists have outliers?

2)  What information does the stem plot show that the histogram hides?

3)  What information does the quantile plot show that the stem plot hides?

Goals:     Be able to use the calculator to make (and be able to interpret) a quantile plot, using the program QUANTILE.  Be able to make a stem plot by hand.

Skills:

…                        Use the TI-83 to create an appropriate histogram or quantile plot.  STAT PLOT and QUANTILE are our two main tools for viewing distributions of data.  Histograms are common displays, but have flaws; the choice of class width is troubling as it is not unique.  The quantile plot is more reliable, but less common.  For interpretation purposes, remember that in a histogram tall boxes represent places with lots of data, while in a quantile plot those same high-density data places are represented by steepness.

…                        Create a stem plot by hand.  The stem plot is a convenient manual display; it is most useful for small datasets, but not all datasets make good stem plots.  Choosing the "stem" and "leaves" to make reasonable displays will require some practice.  Some notes for proper choice of stems: if you have many empty rows, you have too many stems.  Move one column to the left and try again.  If you have too few rows (all the data is on just one or two stems) you have too few stems.  Move to the right one digit and try again.  Some datasets will not give good pictures for any choice of stem, and some benefit from splitting or rounding (see the example on page 13).

…                        Describe shape, center, and spread.  From each of our graphs, you should be able to make general statements about the shape, center, and spread of the distribution of the variable being explored.  One of the main conclusions we want to make about lists of data when we are doing inference (Chapters 6 to 8) is whether the data is close to symmetric; many times "close enough" is, well, close enough!  We will discuss this in more detail when we see the Central Limit Theorem in Chapter 5.

Reading: Section 1.2.

Day 4

Activity: Dance Fever example.  Use the "Arizona Temps" dataset to calculate the mean, the standard deviation, the 5-number summary, and the associated box plot for any of the variables.

Compare these measures with the corresponding histograms and quantile plots you did on Day 2.  Note the similarities (where the data values are dense, and where they are sparse) but especially note the differences.  The box plots and numerical measures cannot describe shape very well.  The histograms are hard to use to compare two lists.  The stem and leaf is difficult to modify.

Answer these questions:

1)  Are high and low temperatures distributed the same way, other than the obvious fact that highs are higher than lows?

2)  How does a single case affect the calculator's routines?  (What if we had had an outlier?)

3)  What information does the box plot disguise?

To calculate our summary statistics, we will use 1-Var Stats (to use List 1) or 1-Var Stats L2 for List 2, for example.  There are two screens of output; we will be mostly concerned with the mean , the standard deviation Sx, and the five-number summary on screen two.

Goals:     Compare numerical measures of center.  Summarize data with numerical measures and box plots.  Compare these new measures with the histograms, stem plots, and quantile plots you made on Day 3.

Skills:

…                        Understand the effect of outliers on the mean.  The mean (or average) is unduly influenced by outlying (unusual) observations.  Therefore, knowing when your distribution is skewed is helpful.

…                        Understand the effect of outliers on the median.  The median is almost completely unaffected by outliers.  For technical reasons, though, the median is not as common in scientific applications as the mean.

…                        Use the TI-83 to calculate summary statistics.  Calculating may be as simple as entering numbers into your calculator and pressing a button.  Or, if you are doing some things by hand, you may have to organize information the correct way, such as listing the numbers from low to high.  On the TI-83, the numerical measures are calculated using STAT CALC 1-Var Stats L#.  Please get used to using the statistical features of your calculator to produce the mean.  While I know you can calculate the mean by simply adding up all the numbers and dividing by the sample size, you will not be in the habit of using the full features of your machine, and later on you will be missing out.

…                        Compare several lists of numbers using box plots.  For two lists, the best simple approach is the back-to-back stem plot.  For more than two lists, I suggest trying box plots, side-by-side, or stacked.  At a glance, then, you can assess which lists have typically larger values or more spread out values, etc.

…                        Understand box plots.  You should know that the box plots for some lists don't tell the interesting part of those lists.  For example, box plots do not describe shape very well; you can only see where the quartiles are.  Alternatively, you should know that the box plot can be a very good first quick look.

Reading: Section 1.2.

Day 5

Activity: Create the following lists:

1)  A list of 10 numbers that has only one number below the mean.

2)  A list of 10 numbers that has the standard deviation greater than the mean.

3)  A list of 10 numbers that has a standard deviation of zero.
For your fourth list start with any 21 numbers.  Find a number N such that 14 of the numbers in your list are within N of the average.  For example, pick a number N (say 4), calculate the average plus 4, the average minus 4, and count how many numbers in your list are between those two values.  If the count is less than 14, try a larger number for N (bigger than 4).  If the count is more than 14, try a smaller number for N (smaller than 4).

Finally, compare the standard deviation to the Inter Quartile Range (IQR = Q3 - Q1).

(You may use any extra time today to discuss Presentation 1 in your groups.)

Goals:     Interpret standard deviation as a measure of spread.

Skills:

…                        Understand standard deviation.  At first, standard deviation will seem foreign to you, but I believe that it will make more sense the more you become familiar with it.  In its simplest terms, the standard deviation is non-negative number that measures how "wide" a dataset is.  One common interpretation is that the range of a dataset is 4 standard deviations.  Another interpretation is that the standard deviation is roughly ¾ times IQR.  Eventually we will use the standard deviation in our calculations for statistical inference; until then, this measure is just another summary statistic, and getting used to this number is your goal.  The normal curve of the next section will further help us understand standard deviation.

Reading: Section 1.3.

Day 6

Activity: Introduce the TI-83's normal calculations.  Homework 1 due.

DISTR normalcdf( lower, upper ) calculates the area under a normal curve between lower and upper.  If you specify just 2 values, mean 0 and standard deviation 1 are assumed.  If you want a different mean or standard deviation, add a third and fourth parameter.  Example: DISTR normalcdf( -10, 20, 5, 10 ) finds the area between -10 and +20 on a normal curve with mean 5 and standard deviation 10 while DISTR normalcdf( -2, 2 )  finds the area on the standard normal curve between -2 and +2.

DISTR invNorm( works backwards, but only gives upper as an answer.  It is also referred to as a percentile.  The 90th percentile is that point at which 90 % of the observations are below.  The syntax is DISTR invNorm( .90 ) or DISTR invNorm( .90, 5, 10 ) ; the first example assumes the standard normal curve and reports the 90th percentile.  The second example uses a mean of 5 and a standard deviation of 10 and also reports the 90th percentile.

Note that if the desired area is above a certain number, you will have to use subtraction or symmetry, as DISTR invNorm( only reports values below, or to the left.

Goals:     Introduce normal curve.  Use TI-83 in place of the standard normal table in the text.

Skills:

…                        Know what a z-score is (standardization).  Sometimes, instead of knowing a variable's actual value, we are only interested in how far above or below average it is.  This information is contained in the z-score.  Negative values indicate a below average observation, while positive values are above average.  If the list follows a normal distribution (the familiar "bell-shaped" curve) then it will be relatively rare to have values below -2 or above +2 (only about 5 % of cases).  Even if the list is not normal, surprisingly the z-score still tends to have few values beyond ±2, although this is not guaranteed.

…                        Using the TI-83 to find areas under the normal curve.  When we have a distribution that can be approximated with the bell-shaped normal curve, we can make accurate statements about frequencies and percentages by knowing just the mean and the standard deviation of the data.  Our TI-83 has 2 functions, DISTR normalcdf( and DISTR invNorm( which allow us to calculate these percentages more easily and more accurately than the table in the text.  We use DISTR normalcdf( when we want the percentage as an answer and we use DISTR invNorm( when we already know the percentage but not the value that gives that percentage.

Reading: Section 1.3.

Day 7

Activity: Practice normal calculations.

1)  Suppose SAT scores are distributed normally with mean 800 and standard deviation 100.  Estimate the chance that a randomly chosen score will be above 720.  Estimate the chance that a randomly chosen score with be between 800 and 900.  The top 20% of scores are above what number?  (This is called the 80th percentile.)

2)  Find the Inter Quartile Range (IQR) for the standard normal (mean 0, standard deviation 1).  Compare this to the standard deviation of 1.

3)  Women aged 20 to 29 have normally distributed heights with mean 64 and standard deviation 2.7.  Men have mean 69.3 with standard deviation 2.8.  What percent of women are taller than the average man, and what percentage of men are taller than the average woman?

4)  Pretend we are manufacturing fruit snacks, and that the average weight in a package is .92 ounces with standard deviation 0.05.  What should we label the net weight on the package so that only 5 % of packages are "underweight"?

5)  Suppose that your average commute time to work is 20 minutes, with standard deviation of 2 minutes.  What time should you leave home to arrive to work on time at 8:00?  (You may have to decide a reasonable value for the chance of being late.)

Goals:     Master normal calculations.  Realize that summarizing using the normal curve is the ultimate reduction in complexity, but only applies to data whose distribution is actually bell-shaped.

Skills:

…                        Memorize 68-95-99.7 rule.  While we do rely on our technology to calculate areas under normal curves, it is convenient to have some of the values committed to memory.  These values can be used as rough guidelines; if precision is required, you should use the TI-83 instead.  I will assume you know these numbers by heart when we encounter the normal numbers again in chapters 5 through 8.

…                        Understand that summarizing with just the mean and standard deviation is a special case.  We have progressed from pictures like histograms and quantile plots to summary statistics like medians, means, and standard deviations to finally summarizing an entire list with just two numbers: the mean and the standard deviation.  However, this last step in our summarization only applies to lists whose distribution resembles the bell-shaped normal curves.  If the data's distribution is skewed, or has any other shape, this level of summarization is insufficient.  Also, it is important to realize that these calculations are only approximations.

…                        Interpret a normal quantile plot.  We often want to know if a list of data can be approximated with a normal curve.  While we might try histograms and quantile plots to see if they "look normal", it is a difficult task, because we have to match the shape to the very special shape of the normal curve.  One simple alternative graphical method is the normal quantile plot.  This plot is nearly identical to a quantile plot, but instead of graphing the percentiles, we graph the z-scores.  Our TI-83 does this for us; the sixth icon in the STAT PLOT Type.  Be cautious though; the graph, as usual, is unlabeled.  However, we only care if the graph is nearly a straight line or not.

Reading: Sections 2.1 and 2.2.

Day 8

Activity: Using the "Arizona Temps" data, plot "Flagstaff High" versus "Phoenix High".

Then guess what the correlation coefficient might be without using your calculator.  Use the sample diagrams on page 126 to guide you.

Finally, using your calculator, calculate the actual value for the correlation coefficient and compare it to your guess.

Repeat for the variables "Flagstaff High" and "Flagstaff Low".

Goals:     Display two variables and measure (and interpret) linear association using the correlation coefficient.

Skills:

…                        Plot data with a scatter plot.  This will be as simple as entering two lists of numbers into your TI-83 and pressing a few buttons, just as for histograms or box plots.  Or, if you are doing plots by hand you will have to first choose an appropriate axis scale and then plot the points.  You should also be able to describe overall patterns in scatter diagrams and suggest tentative models that summarize the main features of the relationship, if any.

…                        Use the TI-83 to calculate the correlation coefficient.  We will have to use the regression function STAT CALC LinReg(ax+b) to calculate correlation, r.  First, you will have to have pressed DiagnosticOn.  Access this command through the CATALOG (2nd 0).  If you type ENTER after the STAT CALC LinReg(ax+b) command, the calculator assumes your lists are in columns L1and L2; otherwise you will type where they are, for example STAT CALC LinReg(ax+b) L2, L3.

…                        Interpret the correlation coefficient.  You should know the range of the correlation coefficient (-1 to +1) and what a "typical" diagram looks like for various values of the correlation coefficient.  Again, page 126 is your guide.  You should recognize some of the things the correlation coefficient does not measure, such as the strength of a non-linear pattern.

Reading: Section 2.2.

Day 9

Activity: Outlier effects on correlation.  The dataset we will explore today has 7 data points.  Plot them and calculate the correlation coefficient.

Add an eighth point in three different places and for each new dataset, recalculate the correlation coefficient.

Summarize the effect of outliers in a paragraph.

(You may use any extra time today to discuss Presentation 1 in your groups.)  Homework 2 due.

Goals:     Understand the impact of outliers on correlation.

Skills:

…                        Interpret the correlation coefficient.  You should recognize how outliers influence the magnitude of the correlation coefficient.  One simple way to observe the effects of outliers is to calculate the correlation coefficient with and without the outlier in the dataset and compare the two values.  If the values vary greatly (this is a judgment call) then you would say the outlier is "influential".

Reading: Section 2.3.

Day 10

Activity: Using the Olympic data, fit a regression line to predict the 2004 and 2008 race results.

Goals:     Practice using regression with the TI-83.  We want the regression equation, the regression line superimposed on the plot, the correlation coefficient, and we want to be able to use the line to predict new values.

Skills:

…                        Fit a line to data.  This may be as simple as 'eyeballing' a straight line to a scatter plot.  However, to be more precise, we will use least squares, STAT CALC LinReg(ax+b) on the TI-83, to calculate the coefficients, and VARS Statistics EQ RegEQ to type the equation in the Y= menu.  You should also be able to sketch a line onto a scatter plot (by hand) by knowing the regression coefficients.

…                        Interpret regression coefficients.  Usually, we want to only interpret slope, and slope is best understood by examining the units involved, such as inches per year or miles per gallon, etc.  Because slope can be thought of as "rise" over "run", we are looking for the ratio of the units involved in our two variables.  More precisely, the slope tells us the change in the response variable for a unit change in the explanatory variable.  We don't typically bother interpreting the intercept, as zero is often outside of the range of experimentation.

…                        Estimate/predict new observations using the regression line.  Once we have calculated a regression equation, we can use it to predict new responses.  The easiest way to use the TI-83 for this is to TRACE on the regression line.  You may need to use up and down arrows to toggle back and forth from the plot to the line.  You may also just use the equation itself by multiplying the new x-value by the slope and adding the intercept.  (This is exactly what TRACE is doing.)

Reading: Section 2.3.

Day 11

Activity: Revisit outliers dataset, adding regression lines.  Plot the data again and calculate the regression line.

Add an eighth point in three different places and for each new dataset, recalculate the regression line.

Summarize the effect of outliers in a paragraph.

Goals:     Practice using regression with the TI-83.  We want the regression equation, the regression line superimposed on the plot, the correlation coefficient, and we want to be able to use the line to predict new values.

Skills:

…                        Understand the limitations and strengths of linear regression.  Quite simply, linear regression should only be used with scatter plots that are roughly linear in nature.  That seems obvious.  However, there is nothing that prevents us from calculating the numbers for any data set we can input into our TI-83's.  We have to realize what our data looks like before we calculate the regression; therefore a scatter plot is essential.  In the presence of outliers and non-linear patterns, we should avoid drawing conclusions from the fitted regression line.

Reading: Sections 2.4 and 2.5.

Day 12

Activity: Correlation/Regression summary.  U. S. population example.  Alternate regression models.  Homework 3 due.

1)  For correlation, the variables can be entered in any order; correlation is a fact about a pair of variables.  For regression, the order the variables are presented matters.  If you reverse the order, you get a different regression line.

2)  We must have numerical variables to calculate correlation.  For categorical variables, we would use contingency tables, but not in this course.

3)  High correlation does not necessarily mean a straight line scatter plot.  U. S. population growth is an example.

4)  Correlation is not resistant; the dataset from Days 9 and 11 showed that the placement of a single point in the scatter plot can greatly influence the value of the correlation and the coefficients in the regression equation.

(You may use any extra time today to discuss Presentation 1 in your groups.)

Goals:     See scatter plots and correlation in practice.  Understand correlations limitations and features.

Skills:

…                        Recognize the proper use of correlation, and know how it is abused.  Correlation measures straight line relationships.  Any departures from that model make the correlation coefficient less reliable as a summary measure. Just as for the standard deviation and the mean, the correlation coefficient is affected by outliers.  Therefore, it is extremely important to be aware of data that is unusual.  Some two-dimensional outliers are hard to detect with summary statistics; scatter plots are a must then.

…                        Understand that there are competing regression models.  We have focused our attention on the linear regression models, but as we see on our TI-83, there are many other potential models.  If you use an alternate model, be sure to plot the fitted line along with the scatter plot.  Our usual measure of fit, r2, will not accurately tell the story.

Reading: Chapters 1 and 2.

Day 13

Activity: Presentation 1.  Graphical and Numerical Summaries, Regression and Correlation

Gather 3 to 5 variables on at least 20 subjects; the source is irrelevant, but knowing the data will help you explain its meaning to us.  Be sure to have at least one numerical and at least one categorical variable.  Demonstrate that you can summarize data graphically and numerically.

Also, pick one of the 50 states.  Predict the population in the year 2010 using a regression function (not necessarily linear though).  Describe how you decided upon your model, and explain how good you think your prediction is.

Reading: Chapters 1 and 2.

Day 14

Activity: Exam 1.  This first exam will cover graphical summaries (pictures), numerical summaries (summary calculations), normal curve calculations (areas under the bell curve), scatter plots, correlation, and regression (two-variable summaries).  Some of the questions will be multiple choice.  Others will require you to show your worked out solution.  Section reviews are an excellent source for studying for the exams.  Don't forget to review your class notes and these on-line notes.

Reading: Section 3.1.

Day 15

Activity: History of polls.

to 1936:  Selection Bias

Literary Digest calls the 1936 election for Alf Landon (remember him), in an electoral college landslide.  The poll was performed by sending postcards to people with telephones, magazine subscribers, car owners, and a few people on lists of registered voters.  The first problem was that the sample was biased toward Republican-leaning voters, who could afford magazine subscriptions and telephones during the Great Depression.  The second problem is the response rate error: they received only 2.3 million responses, for a 23% response rate.  This was compounded by a volunteer error: the respondents are most likely people who wanted change, and Roosevelt is president.  Nevertheless, Franklin Roosevelt, the Democrat, wins in a landslide.  Statistician Jessica Utts (Seeing Through Statistics, Jessica M. Utts, Duxbury Press, Wadsworth Publishing Co., 1996 pages 65-66), who examined issues of the Literary Digest from 1936, says "They were very cocky about George Gallup predicting they would get it wrong.  [Gallup helped make his reputation in polling by correctly calling the race.]  The beauty of something like that is that the winner is eventually known."  [Taken from:  http://whyfiles.org/009poll/fiasco.html]

to 1948:  Quota Sampling

Aided by erroneous polling, newspapers prematurely call the presidential election for challenger Thomas Dewey, leading to the famous photograph of an elatedly re-elected Harry Truman.  The problematic polls use then-popular "quota-sampling" techniques.  In other words, they sought out a certain number of men, a certain number of women, and similarly for blacks, whites, and various income levels.  According to statistician Fritz Scheuren, quota sampling in political polling was abandoned after this debacle in favor of random sampling of the population.  [Taken from:  http://whyfiles.org/009poll/fiasco.html]

to present:  Random Sampling

I will give you the numbers and we can see how the polls have improved since 1936, and if there are still any biases towards either Republicans or Democrats.

Goals:     Introduce sampling.  Identify biases.  Explore why non-random samples are not trustworthy.

Skills:

…                        Understand the issues of bias.  We seek representative samples.  The "easy" ways of sampling, samples of convenience and voluntary response samples, may or may not produce good samples, and because we don't know the chances of subjects being in such samples, they are unreliable sampling methods.  Even when probability methods are used, biases can spoil the results.  Avoiding bias is our chief concern in designing surveys.

…                        Huge samples are not necessary.  One popular misconception about sampling is that if the population is large, then we need a proportionately large sample.  This is just not so.  My favorite counter-example is our method of tasting soup.  To find out if soup tastes acceptable, we mix it up, then sample from it with a spoon.  It doesn't matter to us whether it is a small bowl of soup, or a huge vat, we still use only a spoonful.  The situation is the same for statistical sampling; we use a small "spoon", or sample.  The fundamental requirement though is that the "soup" (our population) is "well mixed" (as in a simple random sample - see Day 17).

Reading: Section 3.1.

Day 16

Activity: Lurking variables exercises.  I have a set of problems from Statistics, by Freedman, Pisani, and Purves, which I think will help us think about looking for alternative explanations than the proffered "attractive" conclusion.  Work on each problem for 5 minutes; we will discuss each of them in detail before the class is over.  Be prepared to defend your explanation.

Goals:     Explore experimentation ideas.  Discover potential lurking variables and ways to control for them.

Skills:

…                        Examine situations and detect lurking variables.  When searching for lurking variables, it is not enough to suggest variables that might also explain the response variable.  Your potential "lurking" variable must also be associated with the explanatory variable.  So, for example, suppose you are trying to explain height using weight.  A possible lurking variable might be age, because age tends to be associated with weight and height.  On the other hand, a variable associated with height that is unlikely to be related to weight (and therefore would not be a lurking variable) is arm span.

…                        Know appropriate ways to attempt to control for lurking variables.  Once a potential lurking variable has been identified, we can make more appropriate comparisons by dividing our subjects into smaller, more homogeneous groups and then making the comparison.

…                        Understand that experimentation, done properly, will allow us to establish cause-and-effect relationships.  Observational studies have lurking variables; we can try to control for them by various methods, but we cannot eliminate them.  If the data is collected appropriately through good experimentation, however, the effects of lurking variables can be eliminated.  This is done through randomization, the thinking being that if a sample is large enough, it can't realistically be the case that all of one group contains all the large values of a lurking variable, for example.

Reading: Section 3.2.

Day 17

Activity: Creating random samples.  We will use three methods of sampling today: dice, Table B in our book, and our calculator.  To make the problem feasible, we will only use a population of size 6.  (I know this is unrealistic in practice, but the point today is to see how randomness works, and hopefully trust that the results extend to larger problems.)  Pretend that the items in our population (perhaps they are people) are labeled 1 through 6.  For each of our methods, you will have to decide in your group what to do with "ties".  Keep in mind the goal of simple random sampling: at each stage, each remaining item has an equal chance to be the next item selected.

By rolling dice, generate a sample of three people.  (Let the number on the die correspond to one of the items.)  Repeat 20 times, giving 20 samples of size 3.

Using Table B, starting at any haphazard location, select three people. (Let the random digit correspond to one of the items.)  Repeat 20 times, giving 20 more samples of size 3.

Using your TI-83, select three people. select three people.  The TI-83 command MATH PRB randInt( 2, 4, 5 ) will produce 5 numbers between 2 and 4, inclusive, for example.  (If you leave off the third number, only one value will be generated.)  Repeat 20 times, giving 20 more samples of size 3.

Your group should have drawn 60 samples at the end.  Keep careful track of which samples you selected; record your results in order, as 125 or 256, for example.  (125 would mean persons 1, 2, and 5 were selected.)  We will pool the results of everyone's work together on the board.

Goals:     Gain practice taking random samples.  Understand what a simple random sample is.  Become familiar with MATH randInt.  Accept that calculator is random.

Skills:

…                        Know the definition of a Simple Random Sample (SRS).  Simple Random Samples can be defined in two ways:
1)  An SRS is a sample where, at each stage, each item has an equal chance to be the next item selected.
2)  A scheme were every possible sample has an equal chance to be the sample results in an SRS.

…                        Select an SRS from a list of items.  The TI-83 command MATH randInt will select numbered items from a list randomly.  If a number selected is already in the list, ignore that number and get a new one.  Remember, as long as each remaining item is equally likely to be chosen as the next item, you have drawn an SRS.

…                        Understand the real world uses of SRS's.  In practice, simple random samples are not that common.  It is just too impractical (or impossible) to have a list of the entire population available.  However, the idea of simple random sampling is essentially the foundation for all the other types of sampling.  In that sense then it is very common.

Reading: Sections 3.3 and 3.4.

Day 18

Activity: Alternate Sampling Schemes.

Using a small population, we will explore alternate sampling schemes.  For each of the methods below, use simple random sampling as appropriate to draw a more complicated random sample.  Then, for your sample, find the average and the standard deviation of the variable of interest.  Use your average to guess the total for the entire population.  Keep track of your results, as we will pool the class results to see the effects of the different methods.

Our population today is the 50 United States.  Our variable of interest is the Land Area.

Simple Random Sampling:  Randomly select 10 states.

Systematic Sampling:  Using the alphabetically sorted list, choose a random number between 1 and 5.  Then select every 10th item after that.  In general, you would use a random number between 1 and N/n, and then select every nth item.

Stratified Sampling:  From the large states (Alaska, Texas, California, Montana, New Mexico, Arizona, Nevada, Colorado, Wyoming, and Oregon), randomly select 4 states.  From the small states (Rhode Island, Delaware, Connecticut, Hawaii, New Jersey, Massachusetts, New Hampshire, Vermont, and Maryland), randomly select 1 state.  From the remaining 31 states, randomly select 5 states.

To guess the total for the entire U. S., guess the totals for the three strata separately.  That is, for the small states, multiply your average for them by 9.  For the large states, multiply by 10.  For the other states, multiply by 31.

Cluster Sampling:  Using the following breakdown of the states (North, Southwest, Central, Southeast, and Northeast), randomly select two regions.  From each region, randomly select 5 states.

North:  Alaska, Washington, Oregon, Nevada, Idaho, Montana, Wyoming, N. Dakota, S. Dakota, Nebraska, and Minnesota.
Southwest:  Hawaii, California, Nevada, Utah, Arizona, Colorado, New Mexico, Kansas, Oklahoma, and Texas.
Central:  Iowa, Missouri, Arkansas, Wisconsin, Michigan, Illinois, Indiana, Ohio, Kentucky, and Tennessee.
Southeast:  Louisiana, Mississippi, Alabama, Georgia, Florida, S. Carolina, N. Carolina, Virginia, W. Virginia, and Maryland.
Northeast:  Pennsylvania, Delaware, New Jersey, New York, Vermont, New Hampshire, Massachusetts, Connecticut, Rhode Island, and Maine.

For our entire class, which method produced the most reliable results?  I will summarize the benefits and drawbacks of each method, despite whether we actually saw these effects in our simulation today.

Homework 4 due.

Goals:     Understand the differences (good and bad) between various sampling schemes.

Skills:

…                        Systematic Sampling.  In systematic sampling, we decide on a fraction of the population we would like to sample, and then randomly select a number between 1 and n, the sample size.  Then from the list of items, we take every N/nth item.  This scheme works best when we have a list of items, and it is simple to select items periodically from that list.  For example, we may have decided to take very 20th name, and there are 20 names per page.  It would be thus very convenient to take the 11th item from each page, for example.

…                        Stratified Sampling.  In stratified random sampling, we draw a simple random sample from several sub-populations, called strata.  The sample size may differ by stratum; in fact, this leads to the most efficient use of stratified sampling.  This scheme works best when there is variability between strata.  For example, one stratum may be more homogeneous than another stratum.  Then, just a few items are needed from the stratum that is homogenous, since all the items in that stratum are basically the same.  Alternatively, from a stratum that is quite diverse, you should sample more heavily.  In our example in class today, the large states are quite different from one another in areas, so we took 40 % of them.  The small states are very similar in area, so we took only 1 (10 % of them).  The remaining 31 states are in between, so we sampled 16 % of them.

…                        Cluster Sampling.  In cluster sampling, we select several groups of items.  Within each selected item, we then repeat by selecting from several smaller groups.  This process continues until we have our ultimate sampled units.  This scheme works best when we do not have a list of all the items in a population, but we have lists of groups of items, for example states or counties.  One drawback of this method is that the ultimate clusters are generally quite similar (all people living on a block in a town are generally more similar than different), so the effective sample size is lower than it appears.  To compensate, cluster samples typically have larger sample sizes.

Reading: Section 4.1.

Day 19

Activity: What is Randomness?  Our notions of probability theory are based on the "long run", but our everyday lives are dominated by "short runs".  Today we will look at some everyday sequences to see if they exhibit this behavior.

Coin experiment 1:  Write down a sequence of H's and T's representing head and tails, pretending you are flipping a coin.  Then flip a real coin 50 times and record these 50 H's and T's.  Without knowing which list is which, in most cases I will be able to identify your real coin.

Baseball players:  In sports you often hear about the "hot hand".  We will pick a player, look at his last 20 games, and see if flipping a coin will produce a simulation that resembles his real performance.  Then we will examine whether we could pick out the simulation without knowing which was which.

Coin experiment 2:  Spin a penny on a flat surface, instead of tossing it into the air.  Record the percentage of heads.

Coin experiment 3:  Balance a nickel on its edge on a flat surface.  Jolt the surface enough so that the nickel falls over, and record the percentage of heads.

Goals:     Observe some real sequences of random experiments.  Develop an intuition about variability.

Skills:

…                        Recognize the feature of randomness.  Random does not mean haphazard, or without pattern.  We cannot predict what will happen on a single toss of a coin, but we can predict what will happen in 1,000 tosses of a coin.  This is the hallmark of a random process: uncertainty in a small number of trials, but a predictable pattern in a large number of trials.

…                        Resist the urge to jump to conclusions with small samples.  Typically our daily activities do not involve large samples of observations.  Therefore our ideas of "long run" probability theory are not applicable.  You need to develop some intuition about when to believe an observed experiment, and when to doubt the results.  We will hone this intuition as we develop our upcoming inference methods.  For now, understand that you may be jumping to conclusions by just believing the simulation's observed value.

Reading: Section 4.2.

Day 20

Activity: Sample Spaces, Simulation.

Using either complete sampling spaces (theory) or simulation, find (or estimate) these chances:

1)  Roll two dice, one colored, one white.  Find the chance of the colored die being less than the white die.

2)  Roll three dice and find the chance that the largest of the three dice is a 6.  (Ignore ties; that is, the largest value when 6, 6, 4 is rolled is 6.)

3)  Roll three dice and find the chance of getting a sum of less than 8.

Goals:     Create sample spaces.  Use simulation to estimate probabilities.

Skills:

…                        List simple sample spaces.  Flipping coins and rolling dice are common events to us, and listing the possible outcomes lets us explore probability distributions.  We will not delve deeply into probability rules; rather, we are more interested in the ideas of probability and I think the best way to accomplish this is by example.

…                        Know the probability rules and how to use them.  We have three rules we use primarily: the complement rule, the addition rule for disjoint events, and the multiplication rule for independent events.  The complement rule is used when the calculation of the complement turns out to be simpler than the even itself.  For example, the complement of "at least one" is "none", which is a simpler event to describe.  The addition rule for disjoint events is used when we are asking about the chances of one event or another occurring.  If the two events are disjoint (they have no elements in common) we can find the chance of their union (one event or the other) by adding their individual probabilities.  The multiplication rule for independent events is used when we have a question about the intersection of two sets, which can be phrased as a question about two events occurring simultaneously.  Both sets occurring at once can be phrased as one event and the other event occurring, so the multiplication rule is used for "and" statements.

…                        Simulation can be used to estimate probabilities, but only for a very large number of trials.  If the number of repetitions of an experiment is large, then the resulting observed frequency of success can be used as an estimate of the true unknown probability of success.  However, a "large" enough number of repetitions may be more than we can reasonably perform.  For example, for problem 1 today, a sample of 100 will give results between 32/100 and 51/100 (.32 to .51) 95% of the time.  That may not be good enough for our purposes.  Even with 500, the range is 187/500 to 230/500 (.374 to .460).  Eventually the answers will converge to a useful percentage; the question is how soon that will occur.  We will have answers to that question after Section 8.1.

Reading: Section 4.3.

Day 21

Activity: Continue coins and dice.  Introduce Random Variables and Probability Histograms.  We will finish up the problems from Day 20, and also examine Pascal's triangle, which is a way of figuring binomial probabilities (chances on fair coins).  Also in our tables, we will include random variables.

Goals:     Understand that variables may have values that are not equally likely.

Skills:

…                        Understand sampling distributions and how to create simple ones.  We have listed sample spaces of equally likely events, like dice and coins. Events can further be grouped together and assigned values.  These new groups of events may not be equally likely, but as long as the rules of probability still hold, we have valid probability distributions.  Pascal's Triangle is one such example, though you should realize that it applies only to fair coins.  We will work with "unfair coins" (proportions) later, in Chapter 8.  Historical note: examining these sampling distributions led to the discovery of the normal curve in the early 1700's.  We will copy their work and "discover" the normal curve for ourselves too using dice.

Reading: Section 4.4.

Day 22

Activity: Means and Variances Rules.  Using the frequency option on STAT CALC 1-Var Stats to calculate mx and sx.  Simulating data to see the rules in action.

While one could simply memorize these rules, I think it might be more instructive to simulate some data and see the rules at work.  So, we are going to reproduce some data very much like the Example 4.27 data on page 303.  Then we will "tinker" with the parameters and see how things change.

To start with, generate some x-values in L1:
MATH PRB randNorm( mx, sx, 300 ) -> L1.
(Use the values in the problem for mx, sx, my, sy, and r.)

You might think we can use a similar command to generate some y-values in L2.  However, this would ignore the correlation in the two variables.  To account for this, we must "borrow" some results from regression.  The next two commands will put "errors" in L3 and y-values in L2.  Trust me, it works.

MATH PRB randNorm( 0, sy * à( 1 - r2 ), 300 ) -> L3
my – r * sy / sx ( mx - L1 ) + L3 -> L2

Plot L1vs L2 to verify that the data does indeed have a correlation of r.  Calculate the means and standard deviations to see that your simulation is close to the assumed values: STAT CALC 1-Var Stats L1 and STAT CALC 1-Var Stats L2.  (You can also do STAT CALC LinReg(ax+b) to get the correlation coefficient.)

Now let's see how the rules work by doing the sum and the difference of the two "SAT scores": L1 + L2 -> L4 and L1 - L2 -> L5.  Check to see if these simulations agree with the theoretical results by finding the means and standard deviations of L4 and L5: STAT CALC 1-Var Stats L4 and STAT CALC 1-Var Stats L5

Now try this again using a different value for r.  (In particular, see what happens when r = 0.  This is the case for independence.)

Goals:     Explore the rules we have for means and variances using one particular simulation of normal data.  Use the TI-83 to calculate the mean and variance for a discrete distribution.

Skills:

…                        Know how to use the TI-83 to calculate the mean and variance for a discrete distribution.  By including a variable of weights or frequencies, the TI-83 will calculate mx and sx for a discrete distribution.  The syntax is STAT CALC 1-Var Stats L1, L2, where the x-values are entered in L1 and the weights (probabilities expressed as counts) are entered in L2.

…                        Understand the rules for sums and differences and linear combinations or random variables.  Through simulation, you should have an intuitive feel for why the correlation has so much to do with the variance of a sum or difference.  In particular, with higher correlations, the variance of the sum increases, while the variance of a difference decreases.  With no correlation, the variance of the sum is the same as the variance of the difference.  Also you should be able to see quite easily why the linear combination rules work.

Reading: Section 4.5.

Day 23

Activity: Constructing probability trees.  Demonstrating Bayes' formula with the rare disease problem.  Homework 5 due.

Consider a card trick where two cards are drawn sequentially off the top of a shuffled deck.  (There are 52 cards in a deck, 4 suits of 13 ranks.)  We want to calculate the chance of getting hearts on the first draw, on the second draw, and on both draws.  We will organize our thoughts into a tree diagram, much like water flowing in a pipe.  On each branch, the label will be the probability of taking that branch; thus at each node, the exiting probabilities (conditional probabilities) add to one. 

On the far right of the tree, we will have the intersection events.  Their probability is found by multiplying.

Calculate the chances of:

1)  Drawing a heart on the first card.
2)  Drawing a heart on the second card.
3)  Drawing at least one heart.
4)  Drawing two hearts.
5)  Drawing a heart on the second draw given that a heart was drawn first.
6)  Drawing a heart on the first draw given that a heart was drawn first.

Now we will do this work for the rare disease problem.

Goals:     Be able to express probability calculations as tree diagrams.  Be able to reverse the events in a probability tree, which is what Bayes' formula is about.

Skills:

…                        Know how to use the multiplication rule in a probability tree.  Each branch of a probability tree is labeled with the conditional probability for that branch.  To calculate the joint probability of a series of branches, we multiply the conditional probabilities together.  Note that at each branching in a tree, the (conditional) probabilities add to one, and that overall, the joint probabilities add to one.

…                        Recognize conditional probability in English statements.  Sometimes the key word is "given".  Other times the conditional phrase has "if".  But sometimes the fact that a statement is conditional is disguised.  For example:  "Assuming John buys the insurance, what is the chance he will come out ahead" is equivalent to "If John buys insurance, what is the chance he will come out ahead".

…                        Be able to use the conditional probability formula to reverse the events in a probability tree.  The key here is the symmetry of the events in the conditional probability formula.  We exchange the roles of A and B, and tie them together with our formula for Pr(A and B).  This reversal is the essence of Bayes' formula.

…                        Know the definition of independence.  Independence is a fact about probability, not about sets.  Contrast this to "disjoint" which is a property of sets.  In particular, independent events are by definition not disjoint.  Independence is important later as an assumption as it allows us to multiply individual probabilities together without having to worry about conditional probability.

Reading: Section 5.1.

Day 24

Activity: Coin flipping is a good way to understand randomness, but because most coins have a probability of heads very close to 50 %, we don't get the true flavor of the binomial distribution.  Today we will simulate the flipping of an unfair coin; that is, a binomial process with probability not equal to 50 %.

Experiment 1:  Our unfair "coin" will be a die, and we will call getting a 6 a success.  Roll your die 10 times and record how many sixes you got.  Repeat this process 10 times each.  Your group should have 40 to 50 trials of 10 die rolls.  Pool your results and enter the data into a list on your calculator.  We want to see the histogram (be sure to make the box width reasonable) and calculate the summary statistics, in particular the mean and variance.  Also produce a quantile plot.  Compare the simulated results with theory.

Experiment 2:  Your calculator will generate binomial random variables for you, but it is not as illuminative as actually producing the raw data yourself.  Still, we can see the way the probability histogram looks (if we generate enough cases; this is an application of the law of large numbers).  I suggest 100 at a minimum.  Again be sure to make your histogram have an appropriate box width.  The command is MATH PRB randBin( n, p, r ), where n is the sample size, p is the probability of success, and r is the number of times to repeat the experiment.

Goals:     Become familiar with the binomial distribution and its applications.

Skills:

…                        Know the four assumptions.  The binomial distribution requires four assumptions:  1)  There must be only two outcomes.  2)  Trials must be independent of one another.  3)  There must be a fixed sample size, chosen ahead of time and not determined while the experiment is running.  4)  The probability of success is the same from trial to trial.  Number 2 is the most difficult to check, so it is usually simply assumed.  Number 4 rules out finite populations, where the success probability depends on what is left in the pool.  Number 3 rules out situations where the experiment continues until some event happens.

…                        Know how to calculate binomial probabilities.  Using DISTR binompdf( n, p, x ), we can calculate the chance of getting exactly x successes in n trials with constant probability p.  Using DISTR binomcdf( n, p, x ), we can calculate the chance of getting at most x successes in n trials with constant probability p.  The second command adds up repeated results of the first command, starting at x and working down to zero.  Because binomcdf only calculates x successes or less, when we want to know about y successes or more, we must use the complement rule of probability.  To find the chance of y successes or more, use y – 1 successes or less.  For example, the chance of 70 or more successes is found this way:  Pr(X Ñ 70) = 1 - Pr(X æ 69).

…                        Mean and Variance.  With algebra, we can show that the mean of a binomial random variable is n p, and the variance is n p (1 - p).  You should memorize these two facts, but don't worry about how they are proven.

…                        Continuity correction.  We can use the normal curve to approximate binomial probabilities.  But because the binomial is a discrete distribution and the normal is continuous, we need to adjust our endpoints by ½ unit.  I recommend drawing a diagram with rectangles to see which way the ½ unit goes.  For example, calculating the chance of getting 40 to 50 heads inclusive on 100 coin flips (perhaps a bent coin) entails using 39.5 to 50.5.

Reading: Section 5.2.

Day 25

Activity: Central Limit Theorem exploration.

In addition to coins and dice, MATH PRB rand on your calculator is another good random mechanism for exploring "sampling distributions".  These examples will give you some different views of sampling distributions.  The important idea is that each time an experiment is performed, a potentially different result occurs.  How these results vary from sample to sample is what we seek.  You are going to produce many samples, and will therefore see how these values vary.

1)  Sums of two items:  Each of you in your group will roll two dice.  Record the sum on the dice.  Repeat this 30 times, generating 30 sums.  Make a histogram and a QUANTILE plot of your 30 sums.  Compare to the graphs of the other members in your group, particularly noting the shape.  Sketch the graphs you made and compare to the theoretical results.

2)  Sums of 4 items:  Each of you generate 4 random numbers on your calculator, add them together, average, and record the result; repeat 30 times.  The full command is: seq ( rand + rand + rand + rand, X, 1, 30 ) / 4 -> L1, which will generate 30 four-sum average random numbers and store them in L1.)  Again, make a graph of the distribution.  (seq is found in the LIST OPS menu.)

3)  Sums of 12 items:  Each of you generate 12 random normal numbers on your calculator using MATH PRB randNorm( 65, 5, 12).  Add them together and record the result; repeat 30 times.  The full command is: seq (sum ( randNorm( 65, 5, 12 ) ), X, 1, 30 ) -> L2.)  Again, make a graph of the distribution.  (This is problem 5.59 in our text.)

For all the lists you generated, calculate the standard deviation and the mean.  We will find these two statistics to be immensely important in our upcoming discussions about inference.  It turns out that these means and standard deviations can be found through formulas instead of having to actually generate repeated samples.  These means depend only on the mean and standard deviation of the original population (the dice or rand or randNorm in this case) and the number of times the dice were rolled or rand was pressed (called the sample size, denoted n).

Goals:     Examine histograms or quantile plots to see that averages are less variable than individual measurements.  Also, the shape of these curves should get closer to the shape of the normal curve as n increases.

Skills:

…                        Understand the concept of sampling variability.  Results vary from sample to sample.  This idea is sampling variability.  We are very much interested in knowing what the likely values of a statistic are, so we focus our energies on describing the sampling distributions.  In today's exercise, you simulated samples, and calculated the variability of your results.  In practice, we only do one sample, but calculate the variability with a formula.  In practice, we also have the Central Limit Theorem, which lets us use the normal curve in many situations to calculate probabilities.

Reading: Section 5.2.

Day 26

Activity: Practice Central Limit Theorem (CLT) problems.  We will have examples of non-normal data and normal data to contrast the diverse cases where the CLT applies.  Homework 6 due.

1)  People staying at a certain convention hotel have a mean weight of 180 pounds with standard deviation 35.  The elevator in the hotel can hold 20 people.  How much weight will it have to handle in most cases?  Do we need to assume weights of people are normally distributed?

2)  Customers at a large grocery store require on average 3 minutes to check out at the cashier, with standard deviation 2.  Because checkout time cannot be negative, they are obviously not normally distributed.  Can we calculate the chance that 85 customers will be handled in a four hour shift?  If so, calculate the chance; if not, what else do you need to know?

3)  Suppose the number of hurricanes in a season has mean 6 and standard deviation à6.  What is the chance that in 30 years there have been fewer than 160 hurricanes?

4)  The number of boys in a 4 child family can be modeled reasonably well with the binomial distribution.  If five such families live on the same street, what is the chance that the total number of boys is 12 or more?


Goals:     Use normal curve with the CLT.

Skills:

…                        Recognize how to use the CLT to answer probability questions concerning sums and averages.  The CLT says that for large sample sizes, the distribution of the sample average is approximately normal, even though the original data in a problem may not be normal.

…                        For small samples, we can only use the normal curve if the actual distribution of the original data is normally distributed.  It is important to realize when original data is not normal, because there is a tendency to use the CLT even for small sample sizes, and this is inappropriate.  When the CLT does apply, though, we are armed with a valuable tool that allows us to estimate probabilities concerning averages.  A particular example is when the data is a count that must be an integer, and there are only a few possible values, such as the number of kids in a family.  Here the normal curve wouldn't help you calculate chances of a family having 3 kids.  However, we could calculate quite accurately the number of kids in 100 such families.

Reading: Chapters 3 to 5.

Day 27

Activity: Catch up day and Review of Probability.

Reading: Chapters 3 to 5.

Day 28

Activity: Presentation 2.  Sampling and Probability (Chapters 3 and 4).

Sample 20 students from UWO.  For each student, record the number of credits they are taking this semester, what year they are in school, and whether or not they are graduating this semester.  Try to make your sample as representative as you can.  You must have a probability sample to get full credit.  Discuss the biases your sample has and what you did to avoid bias.

Your sample has a particular number of men.  Using whatever resources you feel appropriate, calculate and present to us the chance that your sample had the number of males it had.  Be sure you can justify your assumptions.

Reading: Chapters 3 to 5.

Day 29

Activity: Exam 2.  This second exam is on sampling, experiments, and probability, including sampling distributions.  Some of the questions will be multiple choice.  Others will require you to show your worked out solution.  Section reviews are an excellent source for studying for the exams.  Don't forget to review your class notes and these on-line notes.

Day 30

Activity: Guess m&m's percentage.  What fraction of m&m's are blue or green?  Is it 25 %?  33 %?  50 %?  We take samples to find out.

Each of you will sample from my jar of m&m's, and you will all calculate your own confidence interval.  Of course, not everyone will be correct, and in fact, some of us will have "lousy" samples.  But that is the point of the confidence coefficient, as we will see when we jointly interpret our results.

It has been my experience that confidence intervals are easier to understand if we talk about sample proportions instead of sample averages.  Thus I will use techniques from Chapter 8.  Each of you will have a different sample size and a different number of successes.  In this case the sample size, n, is the total number of m&m's you have selected, and the number of successes, x, is the total number of blue or green m&m's in your sample.  Your guess is simply the ratio x/n, or the sample proportion.  We call this estimate p-hat or .  Use STAT TEST 1-PropZInt with 70 % confidence for your interval here today.

When you have calculated your confidence interval, record your result on the board for all to see.  We will jointly inspect these confidence intervals and observe just how many are "correct" and how many are "incorrect".  The percentage of correct intervals should match our chosen level of confidence.  This is in fact what is meant by confidence.

Goals:     Introduce statistical inference - Guessing the parameter.  Construct and interpret a confidence interval.

Skills:

…                        Understand how to interpret confidence intervals.  The calculation of a confidence interval is quite mechanical.  In fact, as we have seen, our calculators do all the work for us.  Our job is then not so much to calculate confidence intervals as it is to be able to understand when one should be used and how best to interpret one.

…                        Know what the confidence level measures.  We would like to always make correct statements, but in light of sampling variability we know this is impossible.  As a compromise, we use methods that work most of the time.  The proportion of times our methods work is expressed as a confidence coefficient.  Thus, a 95 % confidence interval method produces correct statements 95 % of the time.  (By "correct statement" we mean one where the true unknown value is contained in our interval, that is, it is between our two numbers.)

Reading: Section 6.1.  (Skip Bootstrap.  Skip Choosing the Sample Size.)

Day 31

Activity: Changing confidence levels and sample sizes.

Today we will explore how changing confidence levels and sample sizes influence CI's.  Complete the following table, filling in the confidence interval width in the body of the table.  Use STAT TEST 1-PropZInt but in each case make x close to 50 % of n.  (The calculator will not let you use non-integers for x; round off if needed.)


Confidence Level ============> Sample Size

70 %

90 %

95 %

99 %

99.9 %

10

 

 

 

 

 

20

 

 

 

 

 

50

 

 

 

 

 

100

 

 

 

 

 

1000

 

 

 

 

 



We will try to make sense of this chart, keeping in mind the meaning of confidence level, and the desire to have narrow intervals.

Now repeat the above table using STAT TEST ZInterval, with s = 15 and  = 100.

Confidence Level ============> Sample Size

70 %

90 %

95 %

99 %

99.9 %

10

 

 

 

 

 

20

 

 

 

 

 

50

 

 

 

 

 

100

 

 

 

 

 

1000

 

 

 

 

 



Goals:     See how the TI-83 calculates our CI's.  Interpret the effect of differing confidence coefficients and sample sizes.

Skills:

…                        Understand the factors that make confidence intervals believable guesses for the parameter.  The two chief factors that make our confidence intervals believable are the sample size and the confidence coefficient.  The key result is larger confidence makes wider intervals, and larger sample size makes narrower intervals.

…                        Know the details of the Z Interval.  When we know the population standard deviation, s, our method for guessing the true value of the mean, m, is to use a z confidence interval.  This technique is unrealistic in that you must know the true population standard deviation.  In practice, we will estimate this value with the sample standard deviation, s, but a different technique is appropriate (See Day 35).

Reading: Sections 6.2 and 6.4.

Day 32

Activity: Argument by contradiction.  Scientific method.  Type I and Type II error diagram.  Courtroom terminology.  Homework 7 due.

Some terminology:

Null hypothesis.  A statement about a parameter.  The null hypothesis is always an equality or a single claim (like two variables are independent).  We assume the null hypothesis is true in our following calculations, so it is important that the null be a specific value or fact that can be assumed.

Alternative hypothesis.  The alternative hypothesis is a statement that we will believe if the null hypothesis is rejected.  The alternative does not have to be the complement of the null hypothesis.  It just has to be some other statement.  It can be an inequality, and usually is.

One- and Two-Tailed Tests.  A one-tailed test is one where the alternative hypothesis is in only one direction, like "the mean is less than 10".  A two-tailed test is one where the alternative hypothesis is indiscriminate about direction, like "the mean is not equal to 10".  When a researcher has an agenda in mind, he will usually choose a one-tailed test.  When a researcher is unsure of the situation, a two-tailed test is appropriate.

Rejection rule.  To decide between two competing hypotheses, we create a rejection rule.  It's usually as simple as "Reject the null hypothesis if the sample mean is greater than 10.  Otherwise fail to reject."  We always want to phrase our answer as "reject the null hypothesis" or "fail to reject the null hypothesis".  We never want to say "accept the null hypothesis".  The reasoning is this:  Rejecting the null hypothesis means the data have contradicted the assumptions we've made (assuming the null hypothesis was correct); failing to reject the null hypothesis doesn't mean we've proven the null hypothesis is true, but rather that we haven't seen anything to doubt the claim yet.  It could be the case that we just haven't taken a large enough sample yet.

Type I Error.  When we reject the null hypothesis when it is in fact true, we have made a Type I error.  We have made a conscious decision to treat this error as a more important error, so we construct our rejection rule to make this error rare.

Type II Error.  When we fail to reject the null hypothesis, and in fact the alternative hypothesis is the true one, we have made a Type II error.  Because we construct our rejection rule to control the Type I error rate, the Type II error rate is not really under our control; it is more a function of the particular test we have chosen.  The one aspect we can control is the sample size.  Generally, larger sample make the chance of making a Type II error smaller.

Significance level, or size of the test.  The probability of making a Type I error is the significance level.  We also call it the size of the test, and we use the symbol a to represent it.  Because we want the Type I error to be rare, we usually will set a to be a small number, like .05 or .01 or even smaller.  Clearly smaller is better, but the drawback is that the smaller a is, the larger the Type II error becomes.

P-value.  There are two definitions for the P-value.  Definition 1:  The P-value is the alpha level that will cause us to just reject our observed data.  Definition 2:  The P-value is the chance of seeing data as extreme or more extreme than the data actually observed.  Using either definition, we calculate the P-value as an area under a tail in a distribution.  Caution: the P-value calculation will depend on whether we have a one- or a two-tailed test.

Power.  The power of a test is the probability of rejecting the null hypothesis when the alternative hypothesis is true.  We are calculating the chance of making a correct decision.  Because the alternative hypothesis is usually not an equality statement, it is more appropriate to say that power is a function rather than just a single value.

We will examine these ideas using the z-test.  The TI-83 command is STAT TEST ZTest.  The command gives you a menu of items to input.  It assumes your null hypothesis is a statement about a mean m.  you must tell the assumed null value, m0, the alternative claim, either two-sided, or one of the one-sided choices.  You also need to tell the calculator how your information has been stored, either as a list of raw DATA or as summary STATS.  If you choose CALCULATE the machine will simply display the test statistic and the P-value.  If you choose DRAW, the calculator will graph the P-value calculation for you.  You should experiment to see which way you prefer.

Goals:     Introduce statistical inference - Hypothesis testing.

Skills:

…                        Recognize the two types of errors we make.  If we decide to reject a null hypothesis, we might be making a Type I error.  If we fail to reject the null hypothesis, we might be making a Type II error.  If it turns out that the null hypothesis is true, and we reject it because our data looked weird, then we have made a Type I error.  Statisticians have agreed to control this type of error at a specific percentage, usually 5%.  On the other hand, if the alternative hypothesis is true, and we fail to reject the null hypothesis, we have also made a mistake.  This second type of error is generally not controlled by us; the sample size is the determining factor here.

…                        Understand why one error is considered a more serious error.  Because we control the frequency of a Type I error, we feel confident that when we reject the null hypothesis, we have made the right decision.  This is how the scientific method works; researchers usually set up an experiment so that the conclusion they would like to make is the alternative hypothesis.  Then if the null hypothesis (usually the opposite of what they are trying to show) is rejected, there is some confidence in the conclusion.  On the other hand, if we fail to reject the null hypothesis, the most useful conclusion is that we didn't have a large enough sample size to detect a real difference.  We aren't really saying we are confident the null hypothesis is a true statement; rather we are saying it could be true.  Because we cannot control the frequency of this error, it is a less confident statement.

Reading: Section 6.2.

Day 33

Activity: Practice problems on hypothesis testing.  Introduce z-test as an example.

Goals:     Practice contradiction reasoning, the basis of the scientific method.

Skills:

…                        Become familiar with "argument by contradiction".  When researchers are trying to "prove" a treatment is better or that their hypothesized mean is the right one, they will usually choose to assume the opposite as the null hypothesis.  For election polls, they assume the candidate has 50% of the vote, and hope to show that is an incorrect statement.  For showing that a local population differs from, say, a national population, they will typically assume the national average applies to the local population, again with the hope of rejecting that assumption.  In all cases, we formulate the hypotheses before collecting data; therefore, you will never see a sample average in either a null or alternative hypothesis.

…                        Understand why we reject the null hypothesis for small p-values.  The p-value is the probability of seeing a sample result "worse" than the one we actually saw.  In this sense, "worse" means even more evidence against the null hypothesis; more evidence favoring the alternative hypothesis.  If this probability is small, it means either we have observed a rare event, or that we have made an incorrect assumption, namely the null hypothesis.  Statisticians and practitioners have agreed that 5% is a reasonable cutoff between a result that contradicts the null hypothesis and a result that could be argued to be in agreement with the null hypothesis.  Thus, we reject our claim only when the p-value is a small enough number.

Reading: Section 6.2.

Day 34

Activity: Testing Simulation.  In this experiment, you will work in pairs and generate data for your partner to analyze.  Your partner will come up with a conclusion (either reject the null hypothesis or fail to reject the null hypothesis) and you will let them know if they made the right decision or not.  Keep careful track of the success rates.

For each of these simulations, let the null hypothesis mean be 20, n = 10, and s = 5.  You will let m change for each replication.

1)  Without your partner knowing, choose either 16, 18, 20, 22, or 24 for m.  Then use your calculator and generate 10 observations.  Use MATH PRB randNorm( M, 5, 10 ) -> L1 where M is the value of m you chose for this replication.  Clear the screen (so your partner can't see what you did) and give them the calculator.  They will perform a hypothesis test using the .05 significance level and tell you their decision.

2)  Repeat step 1 until you have each done at least 10 hypothesis tests; it is not necessary to have each value of m exactly twice, but try to do each one at least once.  Do m = 20 at least twice each.  (We need more cases for 20 because we're using a small significance level.)

3)  Keep track of the results you got (number of successful decisions and number of unsuccessful decisions) and report them to me so we can all see the combined results.

Goals:     Interpret significance level.  Observe the effects of different values of the population mean.  Recognize limitations to inference.  Realize the potential abuses of hypothesis tests.

Skills:

…                        Interpret significance level.  Our value for rejecting, usually .05, is the percentage of the time that we falsely reject a true null hypothesis.  It does not measure whether we had a random sample; it does not measure whether we have bias in our sample.  It only measures whether random data could look like the observed data.

…                        Understand how the chance of rejecting the null hypothesis changes when the population mean is different than the hypothesized value.  When the population mean is not the hypothesized value, we expect to reject the null hypothesis more often.  This is reasonable, because rejecting a false null hypothesis is a correct decision.  Likewise, when the null hypothesis is in fact true, we hope to seldom decide to reject.  If we have generated enough replications in class, we should see a power curve emerge that tells us how effective our test is for various values of the population mean.

…                        Know the limitations to confidence intervals and hypothesis tests.  Section 6.3 has examples of when our inference techniques are inappropriate.  The main points to watch for are non-random samples, misinterpreting what "rejecting the null hypothesis" means, and misunderstanding what error the margin of error is measuring.  Be sure to read the examples in Section 6.3 carefully as I will not go over them in detail in class.

Reading: Section 6.3.

Day 35

Activity: Gosset Simulation. Homework 8 due.  

Take samples of size 5 from a normal distribution.  Use s instead of s in the standard 95% confidence z-interval.  Repeat 100 times to see if the true coverage is 95%.  (My program GOSSET accomplishes this.)  We will pool our results to see how close we are to 95%.  A century ago, Gosset noticed this phenomenon and guessed what the true distribution should be.  A few years later Sir R. A. Fisher proved that Gosset's guess was correct, and the t distribution was accepted by the statistical community.  Gosset was unable to publish his results under his own name (to protect trade secrets), so he used the pseudonym "A. Student".  You will therefore sometimes see the t distribution referred to as "Student's t distribution".

While we will use the TI-83 to calculate confidence intervals, it will be helpful to know the formulas in addition.  All of the popular confidence intervals are based on adding and subtracting a margin of error to a point estimate.  This estimate is almost always an average, although in the case of proportions it is not immediately clear that it is an average.

Goals:     Introduce t-test.  Understand how the z-test is inappropriate in most small sample situations.

Skills:

…                        Know why using the t-test or the t-interval when s is unknown is appropriate.  When we use s instead of s and do not use the correct t distribution, we find that our confidence intervals are too narrow, and our hypothesis tests reject H0 too often.

…                        Realize that the larger the sample size, the less serious the problem.  When we have larger sample sizes, say 15 to 20, we notice that the simulated success rates are much closer to the theoretical.  Thus the issue of t vs z is a moot point for large samples.

Reading: Section 7.1.

Day 36

Activity: Matched Pairs vs 2-Sample.

Matched Pairs problems are really one sample datasets disguised as two sample datasets because two measurements on the same subject are taken.  Sometimes "subject" is a person; other times it is less recognizable, such as a year.  The key issue is that two measurements have been taken that are related to one another.  One quick way to tell if you have a two sample problem is whether the lists are of different lengths.  Obviously if the lists are of different lengths, they are not paired together.  Naturally the tricky situation is when the lists are of the same length, which occurs often when researchers assign the same number of subjects to each of treatment and control groups.

Once you realize that a sample is a matched pairs data set and that the difference in the two measurements is the important fact, the analysis proceeds just like one sample problems, but you use the list of differences.  In this respect, there is nothing new about the matched pairs situation.

Goals:     Recognize when matched-pairs applies.

Skills:

…                        Detect situations where the matched pairs t-test is appropriate.  The nature of the matched pairs is that each value of one of the variables is associated with a value of the other variable.  The most common example is a repeated measurement on a single individual, like a pre-test and a post-test.  Other situations are natural pairs, like a married couple, or twins.  In all cases, the variable we are really interested in is the difference in the two scores or measurements.  This single difference then makes the matched pairs test a one-variable t-test.

Reading: Section 7.2.

Day 37

Activity: Finish 2-sample work. FIX THIS

Goals:     Complete 2-sample t-test.

Skills:

…                        Know the typical null hypothesis for 2-sample hypothesis tests.  The typical null hypothesis for 2-sample problems, both matched and independent samples, is that of "no difference".  For the matched pairs, we say H0: m=0, and for the 2 independent samples we say H0: m1= m2.  As usual, the null hypothesis is an equality statement, and the alternative is the statement the researcher typically wants to end up concluding.  In both 2-sample procedures, we interpret confidence intervals as ranges for the difference in means, and hypothesis tests as whether the observed difference in means is far from zero.

Reading: Section 8.1.

Day 38

Activity: Proportions: What are the true batting averages of baseball players?  Do we believe results from a few games?  A season?  A career?  We can use the binomial distribution as a model for getting hits in baseball, and examine some data to estimate the true hitting ability of some players.  Keep in mind as we do this the four assumptions of the binomial model, and whether they are truly justifiable.

For a typical baseball player, we can look at confidence intervals for the true percentage of hits he gets.  Using our results from linear combinations (Day 13), we can develop the two sample proportions formulas.  On the calculator, the command is STAT TEST 2-PropZInt.

Technical note:  the Plus 4 Method will give more appropriate confidence intervals.  As this method is extraordinarily easy to use (add 2 to the numerator, and 4 to the denominator), I recommend you always use it when constructing confidence intervals for proportions.  For two sample problems, divide the 2 and 4 evenly between the two samples; that is, add 1 to each numerator and 2 to each denominator.  Furthermore, the Plus 4 Method seems to work even for very small sample sizes, which is not the advice generally given by textbooks for the large sample approximation.  The Plus 4 Method advises that samples as small as 10 will have fairly reliable results; the large sample theory requires 5 to 10 cases in each of the failure and success group.  Thus, at least 20 cases are required, and that is only when p is close to 50 %.

Homework 9 due.

Goals:     Introduce proportions.

Skills:

…                        Detect situations where proportions z-test is correct.  We have several conditions that are necessary for using proportions.  We must have situations where only two outcomes are possible, such as yes/no, success/failure, live/die, Rep/Dem, etc.  We must have independence between trials, which is typically simple to justify; each successive measurement has nothing to do with the previous one.  We must have a constant probability of success from trial to trial.  We call this value p.  And finally we must have a fixed number of trials in mind beforehand; in contrast, some experiments continue until a certain number of successes has occurred.

…                        Know the conditions when the normal approximation is appropriate.  In order to use the normal approximation for proportions, we must have a large enough sample size.  The typical rule of thumb is to make sure there are at least 5 successes and at least 5 failures in the sample.  For example, in a sample of voters, there must be at least 5 Republicans and at least 5 Democrats, if we are estimating the proportion or percentage of Democrats in our population.  (Recall the m&m's example: when you each had fewer than 5 blue or green m&m's, I made you take more until you had at least 5.)

…                        Know the Plus 4 Method.  A recent (1998) result from statistical research suggested that the typical normal theory failed mysteriously in certain unpredictable situations.  Those researchers found a convenient "fix": pretend there are 4 additional observations, 2 successes and 2 failures.  By adding these pretend cases to our real cases, the resulting confidence intervals almost magically capture the true parameter the stated percentage of the time.  Because this "fix" is so simple, it is the recommended approach in all confidence interval problems.  Hypothesis testing procedures remain unchanged.

Reading: Section 8.2.

Day 39

Activity: 2-Sample Proportions

Goals:     2-Sample proportions.

Skills:

…                        Detect situations where the 2-proportion z-test is correct.  Description.

Reading: Chapters 6 to 8.

Day 40

Activity: Review statistical inference,

Goals:     Conclude course topics.  Know everything.

Skills:

…                        Be able to correctly choose the technique from among the z-test, the t-test, the matched pairs t-test, the 2 sample t-test, and z-tests for proportions.  Description.

Reading: Chapters 6 to 8.

Day 41

Activity: Presentation 3.  Statistical Inference (Chapters 6 to 8).  Homework 10 due.

Make a claim, a statistical hypothesis, and test it.  Gather appropriate data to test your claim.  Discuss and justify any assumptions you made.  Explain why your test is the appropriate technique.

Reading: Chapters 6 to 8.

Day 42

Activity: Exam 3.  This last exam covers the z- and t- tests and intervals in Chapters 6 and 7, and the z tests and intervals for proportions in Chapter 8.  Some of the questions will be multiple choice.  Others will require you to show your worked out solution.  Section reviews are an excellent source for studying for the exams.  Don't forget to review your class notes and these on-line notes.

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Last updated December 10, 2006