Exam II Material

1. Exam II Material Probability Distributions, and Parameter Estimation

2. Discrete Distributions • Some Notes • Random Variable – takes on different values based on chance • Discrete – Only has certain possible values • Continuous – Anything is possible!

3. Binomial Distribution • Where only two outcomes are possible • Certain number of “trials” • Trials are independent • Probabilities are consistent

4. Binomial Distribution • On five MC questions with five options, what is the probability that someone randomly guessing will get three correct? • What you’re calculating is: • (# ways to get 3 correct)/(all possible outcomes) • Accounting for the “known” probability • What about three or less correct?

5. Binomial Distribution • Need to find possible ways can occur • Counting Rule for Combinations • Cnx = (n!)/[(x!)(n-x)!] • Tells us number of possible outcomes given situation • Order does not matter

6. Binomial Distributions • With Counting Rule for Combinations and probability, we can construct Binomial Formula • Pr(x) = {(n!)/(x!)(n-x)!}*(px )*(qn-x) • These are located on the Binomial Table

7. Binomial Distributions • Mean • n*p • Gives average number of successes • Standard Deviation • √(n*p*q)

8. Poisson Distribution • Based on a countable number of “successes” • Use Poisson when • We know average number of successes • Probability of success is consistent • Segments are independent • We can divide segments into smaller pieces

9. Poisson Distribution • Mean • λt • Be careful how you use this… • Poisson Probability Disribution • Pr(x) = {(λt)x e-λt}/(x!) • Standard Deviation • √(λt)

10. Poisson Distribution • Rick Ankiel has hit 10 HR in 58 games. What is the probability that he will hit a HR in the first three innings of tonight’s game?

11. Normal Distribution • The most often used/desired distribution of them all • Easiest to work with • Most other distributions converge towards normal • Looking for range of possible values • Pr(x) = 0, no matter what x is • True for all continuous distributions

12. Normal Distribution • Density Function… • Properties of Normal • Has a single peak • Symmetric • Mean = Median = Mode • Approaches 0, but never reaches • Variation depends on height, spread

13. Normal Distribution • All Normals can be “Standardized” • The Z-value is the “standardized” version • This value can be used with the Z-table • But be aware of what you’re calculating and reading from the table

14. Uniform Distribution • Density Function… • Shaped as a rectangle with a and b as its “limits” on the x axis • Mean • (a+b)/2 • Standard Deviation • √{(b-a)2/12}

15. Sampling Distributions • When we consider samples from a population, those samples have a distribution of their own • We’ll want to know how accurate our sample is as a representative of the population

16. Sampling Error • Sampling Error = (x-bar) – μ • Size will depend on sample selection • May be + or – • Can be different for each sample

17. Sampling Distribution • For all possible values of a statistic of a given sample size that has been randomly selected from a population • The average of all possible sample averages will equal population averages • Same is true for standard deviations • This property called unbiasedness

18. Sampling Distributions • As we increase the size of n, something else occurs • As n increases, we should see the values of our statistics (means and standard deviations) grow closer to the population value • This is called consistency • Usually shown analytically as population unknown

19. Sampling Distributions • If population is ~ N, • Sampling dist’n of sample mean ~ N • Mean = μ • Standard Deviation = σ/(√n) • We can then convert to Z-value • Equation…

20. Central Limit Theorem • This is why the Normal is so wonderful • As the sample size grows, any distribution will become approximately normal • Mean of x-bar • Standard Deviation of σ/√n

21. Sampling Dist’n of Proportions • Defined as π = X/N • Sample proportion is p=x/n • Sampling error is p – π • Mean of SampDist of p • π • Standard Error • √{(π(1- π))/n} • Works as long as • nπ ≥ 5 • n(1 – π) ≥ 5

22. Sampling Dist’n of Proportion • We can also do Z-values for this • Z = p – π/(std. error)

23. Estimating Parameters • Point estimate • Statistic used to estimate a parameter • This is likely what you see reported

24. Estimating Parameters, “sigma” known • Recall if the sample is large enough, we can assume it to be normal • Central Limit Theorem • n > 30, typically • Regardless, we can convert to Z-values and construct confidence intervals

25. Estimating Parameters, “sigma” known • Confidence Interval • (X-bar) ± Z*(σ/√n) • This tells you how “certain” you are that the population value is within that range. • The percentage based on choice of Z • Error happens, but it is measurable • Margin of error = Z*(σ/√n) • This illustrates a tradeoff • Lower confidence – lower error • Higher confidence – higher error • Can also increase sample size to lower error

26. Estimating Parameters, “sigma” unknown • We don’t always know σ (in fact, we rarely do) • But we can estimate σ (calculating s) • This however changes our method, slightly • We’ll use the t-distribution • Relying on degrees of freedom

27. Estimating Parameters, “sigma” unknown • t-score for mean… • t-score for confidence interval… • Now we have a method

28. Estimating Parameters • What else could we do to influence the margin of error? • Change the sample size (n)

29. Estimating Parameters • Sample Size Requirement (σ known) • (Z2 σ2)/(e2) • But again, σ not always (if ever) known • Sample Size Requirement (σ unknown) • Estimate σ using (R / 6)

30. Estimating Parameters • We can also do the same for proportions • Some formulas… • Sample Proportion • Standard Error for p • Estimate for SE for p • Confidence Interval for p • Margin of Error • Sample Size

31. Hypothesis Testing • Now that you know how to calculate some statistics, it’s time to “give you the sword” • Null Hypothesis (H0) – This is what we are testing • Alternative Hypothesis (HA) – This includes everything not in the null • One-sided or two sided?

32. Hypothesis testing • Two-sided • H0 will have “=“ • Rejection region is on either side of the null region • One-sided • H0 will have “>” or “<“ • Rejection region is only on one side • When specifying H0, don’t set up the “straw man” • Formally, it goes away from the power of the test • Informally, it’s “shady”

33. Hypothesis testing • What’s the point? • Statistical method of determining validity of claims • Powerful weapon of refuting or supporting these claims • Must be done properly else lose credibility • Note: We will never “prove” anything • We only find evidence

34. Hypothesis Testing • We will either reject or fail to reject H0 • WE WILL NEVER ACCEPT H0!!! • I don’t care what the book says, that is careless and inappropriate

35. Hypothesis Testing • This is done with error • Type I Error – Rejecting a true H0 • Denoted by significance level • This is your α • This will determining your critical value • Type II Error – Failing to reject a false H0 • This is usually denoted by β

36. Hypothesis Testing • To do this, you’ll need your critical value • Critical Value – cutoff point where you either reject or fail to reject H0 • Calculating the critical value… • These will have the subscript “crit” • Critical value compared to the test statistic • Calculating the test statistic… • These will have subscript “stat”

37. Hypothesis Testing • Here’s what you’ll need to do • Specify H0 and HA • Determine if the test is one or two sided • Specify Decision Rule using Zcrit • Calculate Zstat • Compare the two values • Express your decision

38. Hypothesis Testing • Another approach exists • p-value – Tells you what α level would allow you to reject H0 • This does not mean you should use this α • That depends on the problem • Calculating p-value • Find Zstat • Find associated value in Z-table

39. Hypothesis Testing • Once again, σ is not always known • In that event, you’ll again use t-statistics • Calculation of t-stat • Calculation of t-crit • Other than a change in formulas, the procedure is exactly the same

40. Hypothesis Testing • You can also do this for proportions • Calculating… • Z-stat for proportions • Z-crit for proportions