Day By Day Notes for MATH 301

Fall 2006

 

Day 1

Activity: Go over syllabus.  Take roll.  Overview examples: Randomness - coin example.  Gilbert trial.  Election polls. Spam filters.

Creating random samples.  The text is remiss in telling us how to actually select random samples in practice.  Many texts fail in this regard, so to fill in this blank, we will use three methods of sampling today: dice, a table of random digits, 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 trust that hopefully 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 the table of random digits, 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 calculator, select three people.  The TI-83 command MATH 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.)  If your calculator has a rand function only, you can achieve the same result as the TI-83 MATH randInt(2,4) with int(3*rand)+2. 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 items 1, 2, and 5 were selected.)  We will pool the results of everyone's work together on the board.

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

Gain practice taking random samples.  Understand what a simple random sample is.  Become familiar with 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 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.  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 1.1 to 1.6.

 

Day 2

Activity: Dance Fever example.

Use the "Arizona Temps" dataset to calculate means, standard deviations, the 5-number summaries.  To calculate our summary statistics with the TI-83, we will use STAT CALC 1-Var Stats (to use List 1) or STAT CALC 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.

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 5-number summary disguise?

Now, 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 Interquartile Range (IQR = Q3 - Q1).

 

Goals:     Compare numerical measures of center and spread.  Use technology to summarize data with numerical measures.  Interpret standard deviation as a measure of spread.

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 some buttons: STAT CALC 1-Var Stats.  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.  Please get used to using the statistical features of your calculator to produce the means, standard deviations, etc.  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 the boat".

…                        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 about 4 standard deviations.  Another interpretation is that the standard deviation is roughly ¾ times IQR; that is the standard deviation is a bit smaller than the 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 Chapter 6 will further help us understand standard deviation.

Reading: Sections 1.7 to 1.9 and 8.3 (excluding normal quantile-quantile plots).

 

Day 3

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

Compare these measures with the corresponding numerical measures you calculated on Day 2.  Notice that the boxplots 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.

Useful commands for the TI-83:
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.
PRGM EXEC QUANTILE ENTER This program I wrote plots the sorted data and "stacks" them up.  It is essentially a quantile plot.

Using the plots now instead of the summary statistics, answer these questions again:

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 5-number summary disguise?

 

Goals:     Be able to use the calculator to make a histogram, boxplot, or a quantile plot.  Be able to make a stemplot by hand.

Skills:

…                        Summarize data into a frequency table.  The easiest way to make a frequency table is to TRACE the boxes in a histogram 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.  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 experiment by changing the interval width and see what happens to the diagram.

…                        Use the TI-83 to create an appropriate histogram, boxplot, or quantile plot.  STAT PLOT is our main tool 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 steep.

…                        Create a stemplot by hand.  The stemplot is a convenient manual display; it is most useful for small datasets, but not all datasets make good stemplots.  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 in the text).

…                        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.

…                        Compare several lists of numbers using boxplots.  For two lists, the best simple approach is the back-to-back stemplot.  For more than two lists, I suggest trying boxplots, 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 boxplots.  You should know that the boxplots for some lists don't tell the interesting part of those lists.  For example, boxplots do not describe shape very well (apart from rough symmetry); you can only see where the quartiles are.  Alternatively, you should know that the boxplot can be a very good first quick look at a dataset.

Reading: Sections 2.1 to 2.3.

 

Day 4

Activity: Sample Spaces.  Venn Diagrams.  Coins, Dice.  Pascal's Triangle.

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 multiple values; that is, the largest value when 6, 6, 4 is rolled is a 6.)

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

Goals:     Create sample spaces.  Use Venn diagrams to organize sample spaces.  Use simulation to estimate probabilities.

Skills:

…                        Know the definitions of Sample Space, Event, Outcome, etc.  The basic language of probability will be used throughout the course, so it is important for you to be conversant in it.

…                        Be able to use a Venn diagram.  The Venn diagram is a way of partitioning the sample space into mutually exclusive regions.  It can be useful for simply organizing sets, or sometimes is quite useful in understanding proofs (as we will see in the inclusion/exclusion formula on Day 6.)

…                        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 too 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.

…                        Simulation can be used to estimate probabilities.  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 Chapter ?.

…                        Recognize the usefulness and properties of Pascal's Triangle.  Pascal's Triangle is old (known to the Persians and the Chinese in the 11th century) yet is still quite useful.  There are just two rules to construct Pascal's Triangle:  each row begins and ends with a 1, and each entry is the sum of the two entries above it to the left and the right.  From such a simple construction, though, we encounter many relationships: the combination formula, the triangular numbers, the Fibonacci numbers, the powers of 2, among others.  Our chief interest is in the combination formula and its relationship to the binomial distribution.

Reading: Section 2.3.

 

Day 5

Activity: Presentation 1.

Summaries (Chapters 1 and 8.3)

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.

Combinations vs Permutations.

Goals:     Continue exploring Pascal's Triangle and how it relates to counting (permutations and combinations).

Skills:

…                        Know the Permutation and Combination formulas.  When counting the number of ways of choosing items or ordering items, our formulas are nCr and nPr, respectively.  You will need to work enough problems so that you know when to use each of them.  One way to keep them straight is to think of a Combination as a Committee of people, and a Permutation as a Photograph of that committee.  (There are more permutations than combinations for a particular choice of n and r.)  Also don't forget our trick of listing the complete sample space, but only for small problems!

 

Reading: Sections 2.4 and 2.5.

 

Day 6

Activity: Finish Combinations and Permutations.

Arrange the letters in FREDA.  Arrange the letters in FREED.  Arrange the letters in ERRORS.  Arrange the letters in SETTER.

Demonstrate the Inclusion/Exclusion formula with a 3 set Venn diagram.

Use Venn diagrams to "prove":
A = (A«B) » (A«B')
(A»B)' = A'«B'

Basic probability rules:

Probability is a number between 0 and 1, inclusive.
Mutually Exclusive events add when finding the union.
Mutually Exclusive and exhaustive events add to one.

Goals:     Know the rules of probability, including addition, complement, and inclusion/exclusion.  The multiplication rule will be covered on Day 7.

Skills:

…                        Understand the probability rules.  Being adept at probability begins with knowing definitions and knowing basic formulas.  For example, you can't prove things about mutually exclusive sets if you can't recite the definition of mutually exclusive.  Memorize at first; later it becomes "learned", not "memorized".

…                        Relate the rules to sample spaces.  Remember that the rules we're discussing are all based on counting elements in sample spaces.  Sometimes it is helpful to have a few "standard" examples in mind so conjectures or steps in reasoning can be verified.  For example, the inclusion exclusion principle is shown well with the two-dice problem "what is the chance of at least one six?".  Ignoring the intersection makes the probability too large.

…                        Realize how the Venn diagram can help verify results.  The inclusion/exclusion formula is a good example where a Venn diagram can help with the proof or development.  Other examples are DeMorgan's Laws.  For Bayes' formula, on Day 7, the Venn diagram will also be useful.

Reading: Sections 2.6 to 2.8.

 

Day 7

Activity: Constructing probability trees.  Demonstrating Bayes' with the rare disease problem.

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 (Problem 2.128).

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«B).

…                        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: Sections 3.1 and 3.2.

 

Day 8

Activity: Continue coins and dice.  Introduce Random Variables.

We will finish up the problems from Day 4.  Also in our tables, we will include random variables.

Answer the following questions:

1)  What is the chance of getting a sum of 8 on two dice?
2)  What is the chance of getting a sum of 10 on two dice?
3)  What is the chance of getting a sum of x on two dice, where x is between 1 and 13?
4)  What is the chance of getting 10 heads on 20 flips of a fair coin?
5)  How can you get the TI-83 to graph a probability histogram?

Derive a pmf and its cdf.  Use the sum on two dice as an example.  Know how to work back and forth from one to the other.

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

Skills:

…                        Understand discrete random 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 5.  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.

…                        Know the definition of a discrete probability mass function (pmf).  If a non-negative function sums to 1 over some set, then we have a discrete pmf.  It is not necessary for the set to be finite; this means we may need to work with infinite sums.  Because each item in the sum is a probability, it is necessary that each value is less than one.  (Contrast this with the continuous distributions on Day 9.)

…                        Know the definition of a discrete cumulative distribution function (cdf).  If a non-decreasing function begins at 0 from the left and ends at 1 on the right, and has no place where the derivative is non-zero, then we have a discrete cdf.  The key is that discrete cdf's are stairs, flat spot with discrete jumps.

Reading: Section 3.3.

 

Day 9

Activity: Presentation 2.

Probability (Chapter 2)

Choose one of the following games and
1)  Give us a short history of the game.
2)  Describe how randomness is part of the game.
3)  Using our probability rules, show us an example using this game.

Games:  Risk, Blackjack, Backgammon, Roulette, Battleship, Poker, Minesweeper, Cribbage


Calculate a pmf and its cdf.  Use the uniform as an example.  Know how to work back and forth from one to the other.  Note:  calculus required!

Go over the cdf method of generating random samples.  Requires the cdf in a formula that can be inverted.

Goals:     Introduce continuous distributions.

Skills:

…                        Know the definition of a probability density function (pdf).  If a non-negative function integrates to 1 over some interval, then we have a probability density function.  Notice that the function can certainly be over 1 (contrast to pmf's); the key here is that the area is one, not the maximum height.

…                        Know the definition of a continuous cumulative distribution function (cdf).  If a continuous non-decreasing function begins at 0 from the left and ends at 1 on the right, then we have a discrete cdf.  The key is the continuity.  If a function has only jumps, it is discrete.  If a function has no jumps, it is continuous.  A function with both is mixed.  An example of a mixed distribution is a question like "If you are employed, what is your income?"  People without a job have no income, so there is a spike at 0.

…                        Realize that these formulas and functions we are exploring are simply models.  What we are trying to do with these functions is model real world data with simple curves.  We hope to have both simple equations and close fitting histograms and quantile plots.  The best situation is a model that has parameters that can be "tuned".  By choosing the parameters judiciously we can find curves that approximate real data.

…                        Know how to use the cdf equation to generate random values from the variable.  The cdf ranges from 0 to 1, and each value represents a percentile.  If we produce a uniform number between 0 and 1, such as rand, then by cross-referencing on the graph, we can tell which x-value corresponds to that uniform random number.  While this can be done visually, it's not very useful unless the equation can be solved explicitly.

 

Reading: Section 3.4.

 

Day 10

Activity: Joint Distributions.

Because Calculus III is not a prerequisite for MATH 301, we will not do double integrals or partial derivatives.  However, we can do marginal distributions and explore the discrete problems, as they do not require multiple integrals.

To calculate a marginal distribution, we will integrate over the other variable.  So, the marginal distribution with respect to x is found by integrating out y.  The variable x is treated as a constant.

After we have the marginal, we can use Bayes' formula and find the conditional distribution.  This leads to a notion of independence just as in probability.  This is the key result from this section:  If two distributions are independent, then they're joint pdf is the product of their marginals.

We will finish today with an exploration of the discrete cdf.  The marginal distributions are found the same way as the continuous; we just "add" out the other variable, much like integrating.  The cdf, however, is harder to work with.  In fact, the discrete joint cdf is seldom used, except abstractly in theorems.

Goals:     Extend the notions of pmf, pdf, and cdf to two dimensions.

Skills:

…                        Know what the marginal distributions are and how to derive them from a cdf.  The marginal distributions are the result of "getting rid of" the other variable.  For discrete distributions, we sum; for continuous distributions, we integrate.  One way to visualize this is to imagine looking sideways at a joint pdf or pmf and "smushing" it together in one direction, in essence averaging it, to make it only one-dimensional.

…                        Know the independence definition.  Our main conclusion of interest from this section is that when two distributions are independent, we can get their joint pdf by multiplying the marginals together.  This relationship extends to expected values, so that we can calculate the joint expected value by splitting the sum or integral into two one-variable parts and multiplying the results together.  If the two variables are dependent, however, things are much more complicated, and we won't go there in this course.

…                        Understand the relationship between the cumulative cdf and the pdf for continuous distributions.  Warning:  this requires Calc III.  To find the pdf from the cdf, we take partial derivatives.  To find the cdf from the pdf, we do a cumulative double integral.  I'm not expecting you to perform these operations; rather I want you just to know what operation is required.

…                        Understand why the discrete cdf and discrete pmf are harder to work with in joint distributions than the continuous cdf and pdf.  Due to the stair-step nature of discrete distributions, the pmf is not as straightforward to calculate.  In most cases, there isn't a nice neat formula for the cdf.  The important thing to know is definitions.  The "jumps" do not represent the probability at that point.  (Which jump would one use?  There are three to choose from.)

Reading: Sections 4.1 and 4.2.

 

Day 11

Activity: Expected Value, Variance.  Using the frequency option on STAT CALC 1-Var Stats to calculate m and s for a discrete distribution.

GPA is an EV.  Go over an EV calculation.  Then move on to a function, like x2.  Finally, use variance as the function.

The TI-83 will help us calculate EV and Variance for a discrete distribution.  We need to include a column of weights, integers, which are proportional to the probabilities of the x-values.  Then we use STAT CALC 1-Var Stats L1, L2 where L1 contains the x-values and L2 contains the weights.

Goals:     Define Expected Value and Variance for distributions.

Skills:

…                        Know the definition of Expected Value.  Expected Value is calculated as the average product of the random variable and its probabilities.  For discrete variables, this is a sum, and for continuous variables it is an integral.

…                        Realize that the definition is really the same for both discrete and continuous distributions.  Both definitions involve a summed product.  The nature of the summing is the only difference between them.  Once you realize that integration is simply repeated addition, the similarity of the two formulas is clear.

…                        Know the definition of Variance.  Variance is a particular kind of Expected Value.  The Expected Value of a function of x is simply the average product of that function of x and its probabilities.  For Variance, the function of x that we use is the squared difference from the mean, or Expected Value.

…                        Understand that Expected Values and Variances are parameters and should have no x-values.  After summing or integrating, the variable of summation or integration disappears.  This is most easily seen in the Fundamental Theorem of Calculus where we substitute numbers (the integrands) into the anti-derivative.  Explicitly we note that x equals those integrand values.  Therefore, Expected Value and Variance are constants.

…                        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 m and s 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 integers) are entered in L2.

Reading: Chapters 1 to 3.

 

 Day 12

Activity: Exam 1.

This first exam will cover graphical summaries (pictures), numerical summaries (summary calculations) and probability (including random variables and joint distributions), but excluding Expected Values and Variances from Day 11 (Chapter 4).

Reading: Sections 4.3 and 5.1 to 5.3.

Day 13

Activity: Simulating data to see the Linear Combinations formulas in action.  Introduce Binomial.

Activity 1

Rules for Linear Combinations.  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 SAT data.  Suppose for a recent year, the verbal SAT had mean 507 with standard deviation 111, math SAT had mean 519 with standard deviation 115, and the correlation was 0.71.  We will "tinker" with these 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.)

Activity 2

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 you 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 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.

Activity 3

Using DISTR binompdf( and/or DISTR binomcdf(, calculate these probabilities:

1)  What is the chance of 4 or fewer successes with n = 10 and p = .4?
2)  What is the chance of 3 to 10 successes, inclusive, with n = 20 and p = .7?
3)  What is the chance of more than 6 successes with n = 15 and p = .2?

Goals:     Finish Expectation Formulas.  Introduce the important Binomial distribution.

Skills:

…                        Know how to combine Expected Value and Variance for Linear Combinations of Random Variables.  We often are interested in linear functions of two random variables.  For instance, we may know the mean and variance of times for workers traveling to a job site.  We may also know the mean and variance of the length of the job.  What we really want to know is the total length of time the workers are gone from the main plant.

…                        Know the four assumptions underlying the binomial model.  We must have:
1)  an experiment that has only two outcomes: success and failure.
2)  the trials are independent of one another.
3)  a fixed sample size n.
4)  a constant probability of success p.

…                        Become familiar with the TI-83 commands to calculate binomial probablities.  Our TI-83 has two functions,  and  which allow us to calculate binomial probablities.  As their names imply, one is cumulative and the other is marginal.  With sufficient memory, one could resort to the actual formula and use a sequence command to accomplish the same thing.  In particular, you can verify that you are using the commands correctly by memorizing a small check example.  The command syntax is DISTR binompdf( n, p ), which will give the whole pmf as a list, or DISTR binompdf( n, p, x ), which will give just the probability of x.   For DISTR binomcdf( n, p, x ),  we have the cumulative probability below x.

…                        Random values can be generated using MATH PRB randBin(.  The usual cdf method of generating random values is useless here because we don't have an explicite invertible formula for the binomial cdf.  Thus we rely on other mechanisms.  One could generate a series of Bernoulli trials and add the 0's and 1's together.  The easiest way, however, 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 values required.

Reading: Sections 5.3 to 5.4.

 

Day 14

Activity: Finish Binomial.  Introduce Hypergeometric.

Our last item of business with the binomial is the mean and variance calculation.  I will go through the "trick" in class, which is just a few algebra steps, but it might throw some of you.  I suggest paying attention carefully, and trying to focus on the big picture instead of the details.  The later, knowing what the purpose of each step is you can pay attention to the details.  This trick will appear in other distributions, so it is worth knowing about.

My activity:  All hypergeometric problems can be thought of as "lottery" problems.  The winning balls are our "successes"; the losing balls are the "failures".  In this respect, the hypergeometric is like the binomial.  However, balls are chosen without replacement, and this is where the two distributions differ.  I will do three types of examples:  quality control, poker, and a large binomial.

Your activity:  Is my program  generating hypergeometric data? You will check to see if it is close to theory.  Warning: this will be difficult to answer unequivocally, due to the "Law of Small Numbers".  In your groups, generate a number of observations with the program, using some parameters you choose, and then decide some devices (graphs, calculations, etc) to compare the program output with theory.  At the end of the period, we will share notes, so be prepared in your group to summarize your findings.

Goals:     Understand the Hypergeometric distribution, including generating cases, and the mean and variance.

Skills:

…                        Know the mean and variance for the binomial.  For each distribution we encounter, we will want to know the mean and variance.  These become useful later when we examine the Central Limit Theorem.  For the binomial, the mean is n p and the variance is n p ( 1 - p ).

…                        Recognize situations where the hypergeometric distribution is an appropriate model.  The example I think of when imagining the hypergeometric distribution is the decision to accept or reject a shipment of goods.  Another good example is the full house calculation for poker.  In any case, what we are modeling is a binomial-like situation, with successes and failures.  But see the next item.

…                        Know how the binomial and the hypergeometric are related.  The key difference is the hypergeometric comes from a finite population.  If one imagines balls in urns, then the binomial is a hypergeometric using a huge urn.  (Of course, we're talking about the limit here.)

…                        Know the mean and variance for the hypergeometric.  Because of the close relationship with the binomial, we can use the corresponding formulas, with a slight modification.  If we let p = k / N, and if we use the finite population correction factor ( N - n ) / ( N - 1 ), we see the mean and variance are easy to remember.  For the hypergeometric distribution, the mean is n k / N and the variance is ( N - n ) / ( N - 1 ) n k / N ( 1 - k / N ).

…                        Use  to generate random values from a hypergeometric distribution.  The program HYPER will generate random data which follows the hypergeometric distribution.  The program is simple; it creates a cdf and compares it to a random number MATH PRB rand until the cdf exceeds MATH PRB rand.  That value is the one reported.  In the program, M is the population size (N), K is the number of successes in the population (k), N is the sample size (n), and R is the number of random numbers desired.

Reading: Sections 5.5 and 5.6 and 6.1 to 6.3.

 

Day 15

Activity: Negative Binomial and Poisson.  Introduce the TI-83's normal calculations.

I will derive the pmf for the Negative Binomial by a counting argument.  This will be one of those few times where I think the derivation of a distribution is one you should memorize.  If you know how the formula is developed, you will know the pmf without having to memorize a formula.  The essence of this proof is filling in the slots of a sequence of successes and failures, but requiring the last one to be a success.  It is then just a matter of using the right binomial coefficient and multiplying by the right number of p's and ( 1 - p )'s.

How can we generate random data from this distribution?  Will a program similar to HYPER work?  Or does the infinite support worry us?  Can we simply flip coins (with our calculator)?  After we struggle with this a bit, I will share my two programs I wrote.

The poisson distribution is our last specific discrete model.  It is based on the Taylor's series expansion of ex.  Basically, any series you can invent that adds to a constant can be made into a pmf.


To begin the continuous distributions, we will look at the famous bell-shaped curve, the normal curve.  We have encountered this before, on Day 13, and we see the shape begin to appear if we graph the values from Pascal's Triangle.  Because the pdf for the normal is an equation for which there is no corresponding anti-derivative (in a closed form), we must resort to numerical integration to calculate areas under the curve.  Every statistics textbook has a normal table for this purpose.  You are welcome to use the text's normal table; however, I think you will find it much, much easier to use the built-in functions in the TI-83.

The two commands are DISTR normalcdf( and DISTR invNorm(. DISTR normalcdf(  is used to find the area under the curve between two x-values, and DISTR invNorm( is used to find the percentile.  The command DISTR normalcdf( 10, 20, 14, 3 ) will give the area between 10 and 20 using a mean of 14 and a standard deviation of 3.  If you leave off the mean and standard deviation, 0 and 1 will be assumed.  The command DISTR invNorm( .7, 14, 3) will give the x-value that has area .7 to the left of it, using a mean of 14 and a standard deviation of 3.  Again, if you leave off the mean and standard deviation, 0 and 1 are assumed.

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

Skills:

…                        Relate the Negative Binomial to the Binomial and the Hypergeometric.  The negative binomial differs from the binomial because the number of trials isn't known beforehand; we continue the experiment until r successes occurs.  The negative binomial differs from the hypergeometric because the probability of a success on any trial, p, is fixed, as in the binomial.

…                        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: Sections 6.3 and 6.4.

 

Day 16

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 Interquartile 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 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.

…                        Understand that summarizing with just the mean and standard deviation is a special case.  We have progressed from pictures like histograms to summary statistics like medians, means, etc. to finally summarizing an entire list with just 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 incomplete.  Also, it is important to realize that these calculations are only approximations.

Reading: Sections 6.5 to 6.7.

 

Day 17

Activity: Normal Approximation to the Binomial.  Gamma Distribution.  Cauchy Distribution.

Calculate the chance of getting between 40 and 50 heads on 100 tosses of a fair coin.  Use the binomial.  Then use the normal approximation.

Now try the chance of exactly 50 heads on 100 tosses.

Now redo the first one using the continuity correction.

The next pdf we will explore is the flexible Gamma Distribution.  Unfortunately, calculating areas requires numerical integration, and tables are not readily available.  Our book has a table called the incomplete gamma distribution, but we can use our TI-83's chi-square distribution and some theory from MATH 401 to find these same areas.  Quite simply, convert the desired x-value via y = 2 x / b and use 2 a degrees of freedom, df.  Then DISTR c2cdf( 0, y, df ) on the TI-83 will give the probability that the gamma variable is less than x.  To find right tail areas (above x), we use symmetry (subtraction).  Also, if we know the pdf, we can simply ask our calculators to do a numerical integration to get the desired probability.

The last pdf is a counter-example to some theorems.  The Cauchy Distribution (cdf shaped like arc tangent) is most easily explored by the ratio of two standard normal numbers.  It has unusual behavior though.  While its pdf is symmetric, it has no mean.  This is a curiosity of calculus results; while it surely has a median, the "center of mass" integral is actually undefined.  This behavior is apparent after a long string of outcomes.  Now and then, extreme outliers occur.  Even worse, if you keep a cumulative average, it will not seem to approach any fixed value; it wanders around directionless.  This is in direct contrast to the results of the Central Limit Theorem (Days 18 and 19) because the Cauchy doesn't satisfy one of the hypotheses.  One group will further examine this distribution in the next presentation (Day 22).

Goals:     Understand the details of using the normal curve to approximate binomial probabilities.  Examine Gamma and Cauchy distributions.

Skills:

…                        Know why we add or subtract ½ before doing the normal approximation to the binomial.  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.

…                        Know the shapes of the gamma distribution.  By adjusting the parameters, the gamma distribution can take on a number of shapes.  You should explore the various combinations to see when the curve begins at 0, when it begins at infinity, and when it begins at some value in between.

…                        Know the properties of the gamma distribution.  You should know the mean and variance of the gamma distribution.  the mean is a b and the variance is a b 2.

…                        Know how to generate gamma probabilities when a is an integer.  With the appropriate transformation ( 2 x / b ) we can use our TI-83's c2 function to find gamma probabilities.  If a is not an integer, you will have to interpolate.  Another solution is to use the calculator's definite integral abilities.  This of course requires you to have memorized the pdf.

Reading: Sections 8.1 to 8.5.

 

Day 18

Activity: Central Limit Theorem exploration.

In addition to coins and dice, 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 or a QUANTILE of your 30 sums.  Compare to the graphs of the other members in your group, particularly noting the shape.  Sketch the graph you made and compare to the .

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.

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 ? 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 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: Chapters 4 to 6.

 

Day 19

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.

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: Sections 9.1 to 9.5.

 

Day 20

Activity: Exam 2.

This second exam is on expected value, variance, and particular distributions, both discrete and continuous.  You should know facts about each distribution we have encountered.  You should be able to generate random data from each distribution.  You should know the strengths and limitations of each of them.

Reading: Sections 8.4 and 8.5.

 

Day 21

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 Section 9.10.  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.

Now 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:     Introduce statistical inference - Guessing the parameter.  Construct and interpret a confidence interval. See how the TI-83 calculates our CI's.  Interpret the effect of differing confidence coefficients and sample sizes.

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.

…                        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 22).

Reading: Section 8.7.

 

Day 22

Activity: Presentation 3.

Central Limit Theorem (Section 8.3)

Choose from among the following distributions:  Exponential, Normal, Cauchy CAUTION: this distribution violates the CLT hypotheses, Uniform, Bernoulli, Problem 4.53's distribution.

Generate a sample of 5 items.  Calculate the mean and record the result.  Repeat several hundred times.  Finally, graph the quantile plot of the list of means from your several hundred samples.

Repeat for a sample of 50 items.  Repeat for a sample of 200 items.  Compare all of your results to the theory from the CLT.


Gosset Simulation.  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".

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: Sections 9.8 to 9.11.

 

Day 23

Activity: Matched Pairs vs 2-Sample.  Proportions.

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.

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 1-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 %.

Goals:     Recognize when matched-pairs applies.  Introduce proportions.

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.

…                        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: Sections 10.1 to 10.4.

 

Day 24

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

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  , 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 hyothesis, 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.

…                        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 number.

Reading: Sections 10.5 to 10.7.

 

Day 25

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 H0: m = 20, let n = 10, and let s = 5.  You will change m for each repetition.

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.


In addition to the simulation above, which uses the TI-83's built in routines, you should also be able to calculate the observed probabilities we saw in our chart exactly.  For example, what is the probability that the sample average will cause you to reject the null hypothesis that the mean is 20 when the true mean is really 24?  These calculations are known as power calculations, and are a vital part of the choice of a test or the design of an experiment.  For if you only had a very small chance of detecting an important difference, then the procedure was really a waste of resources; you weren't going to reject anyway.  Typically, these explorations will become sample size problems.

One last point today is that these techniques all require random samples, and typically either normally distributed data or large sample sizes (to invoke the CLT).  In particular be cautious of situations where the entire population has been measured.  In these cases, asking the statistical question "Is the difference due to chance?" makes no sense.  Because the entire population was measure, there is no chance involved; there is no sampling error due to items not being chosen for the sample.  This situation occurs often in science.  For example, a psychologist may conduct an experiment on a litter of rats.  This litter is of course not a random sample of all rats; rather it is the rats available to that researcher at that time.  What is typically assumed then is that this sample of rats is like a random sample.  Whether this assumption is true or not cannot be checked.  Most researchers simply proceed.  I think it is important that you at least acknowledge this "fudging".  In practice, though, there will be little you will do about it.

Goals:     Interpret significance level.  Observe the effects of different values of the population mean.  Recognize limitations to inference.

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.  Often users of statistical inference techniques will blindly use them without checking the implicit hypotheses.  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.

Reading: Section 10.8 and 10.11 to 10.12.

 

Day 26

Activity: Finish 2-sample work.

Proportions. 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.

…                        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 tests for proportions.  When we do not know the population standard deviation, we must use the t-procedures. When we have two dependent samples, we use the matched pairs t-test. When we are working with binomial counts, we use the proportions tests. See the previous days for more lengthy descriptions.

…                        Know the mechanics of the proportions z-tests.  For proportions tests, we use counts of successes and total sample sizes. These are put into a z-formula because in the binomial, once the proportion is known, so is the variance. The resulting test statistic has an approximate normal distribution, for large sample size.

…                        Detect situations where the 2-proportion z-test is correct.  When we have two independent binomial samples, the two-sample proportions test or interval is appropriate.

Reading: Chapters 8 to 10.

 

Day 27

Activity: Presentation 4.

Statistical Inference (Chapters 8 to 10)

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.

Review

Goals:     Conclude course topics.  Know everything.

Reading: Chapters 8 to 10.

 

Day 28

Activity: Exam 3.

This last exam covers the t-tests and intervals and the z-tests and intervals for proportions in Chapters 8 through 10.

 

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Last updated November 26, 2006