Hypothesis Testing: The One Sample Case

1. Hypothesis Testing: The One Sample Case

2. How do we make inference if any sample mean is possible for any µ? Tests of a single mean tell you if a sample mean is likely given a hypothesized µ

3. What operational hypothesis would we consider for this research proposition? • Black Americans continue to experience considerable disadvantages in the labor market, even more than a quarter of a century after Congress enacted major legislation aimed at equalizing opportunity in the workplace

4. Research proposition: American blacks still have an earnings disadvantage • Operational hypothesis: mean income for blacks is lower than for the population as a whole • We draw a random sample of 166 blacks and calculate their mean income for 1990 • Based on census information, we know the mean income for all Americans in 1990 • By comparing the means, we are asking whether it is reasonable to consider the sample of black Americans a random sample that is representative of the population of full-time workers in the U.S.

5. The black sample mean is $5,729 below the population mean, but is it enough below to warrant the conclusion that blacks are not a random sample of the population?

6. What are the possibilities? • 1) The average earnings of the black population are about the same as the national average and this sample happens to show a really low mean difference=sampling variability • 2) The average earnings of the black population are indeed lower than average earnings nationallydifference=real

7. How do we decide which explanation makes more sense? • Answer - We use hypothesis tests • Hypothesis tests: A decision making tool that enables us to evaluate hypotheses about population parameters on the basis of sample statistics

8. Five Step Model for Hypothesis Tests • 1) Making assumptions • 2) Stating hypotheses • 3) Selecting sampling distribution and establish the critical region • 4) Compute the test statistic • 5) Make a decision

9. One Sample Hypothesis Test: Question • A random sample of 166 blacks earned an average of $18,037 in 1990. The national average was $23,766 with a standard deviation of $16,101. Do blacks make significantly less than the population? Set  = .05

10. Step One: Making Assumptions • Hypothesis testing involves several assumptions that must be met for the results of the test to be valid • For the one sample hypothesis test, we assume random sampling, the level of measurement is interval-ratio, and the sampling distribution is normal

11. When can you assume the data are interval-ratio? • You don’t really assume, you check

12. When can you assume the sample was random? • When you know how it was collected (in other words, you check again)

13. When can you assume the shape of the sampling distribution is normal? • When the population distribution is normal • When N is large • This is what we learn from the Central Limit Theorem and its corollary

14. Step Two: Stating Hypotheses • Null Hypothesis: A statement of ‘no difference’ • Research or Alternative Hypothesis: A statement that reflects the research question • Both are expressed in terms of population parameters

15. Step Two: Stating Hypotheses

16. Step Two: Stating Hypotheses

17. Step Three: Select the sampling distribution and establish the critical region • Select Sampling Distribution: Use Z distribution when the sample is large • Critical Region: Area under the sampling distribution that includes all unlikely sample results • Zcritical: Z score associated with a particular  level and marking the beginning of the critical region

18. Step Three: Selecting sampling distribution and establish the critical region • Sampling distribution = Z distribution •  = .05 • Zcritical = - 1.65

19. Step Four: Compute the test statistic

20. Step Four: Compute the test statistic

21. Step Five: Make a Decision • Plot the test statistic on the sampling distribution • If the test statistic is in the critical region, our decision is: reject the null (“statistically significant”) • If the test statistic is not in the critical region, our decision is: fail to reject the null

22. Step Five: Make a Decision • We reject the null hypothesis. We are 95% confident that blacks have significantly lower annual earnings than the population as a whole.

23. One Sample Hypothesis Test: Question • We took a random sample of white women (N = 240), and calculated their mean annual income for 1990 to be $16,653. We know that the annual earnings for the population is $23,766 with a standard deviation of 16,101. Is there a statistically significant difference between white women’s annual earnings and the entire population? Set  = .05

25. Step One • We assume: random sampling, level of measurement is interval ratio, and that the sampling distribution is normal

26. Step Two

27. Step Three • Sampling Distribution = Z distribution •  = .05 • Zcritical = +/- 1.96

28. Step Four

29. Step Five • We reject the null hypothesis. We are 95% confident that white women have significantly different annual earnings than the population as a whole.

30. Available Sampling Distributions for Hypothesis Testing • Use the Z distribution when sample size is large • If sample size is small, we use the t distribution • t distribution = flatter than the normal curve with fatter tails • As the sample size becomes closer to 100, the t distribution begins to approximate the Z distribution

31. One Sample Hypothesis Test: Question • Faculty salary equity is a hot political issue: some argue that earnings of unionized faculty are considerably higher than the earnings of faculty nationwide. Assume that we know the national mean salary of a full-time faculty member is $45,000 and that salaries are normally distributed. We take a random sample of 23 unionized faculty members and find that they make on average $47,000 with a standard deviation of $13,200. Do unionized faculty members make significantly more a year than all faculty members in the population? Set  = .01

33. Step One • We assume random sampling, level of measurement is interval-ratio, and the sampling distribution is normal

34. Step Two

35. Step Three • Sampling Distribution = t distribution •  = .01 • df = N – 1 • df = 22 • tcritical = 2.508

36. Step Four

37. Step Four

38. Step Five • We fail to reject the null hypothesis.

39. One Sample Hypothesis Test • We know that all high school seniors apply to an average of 4.4 colleges. We took a random sample of 25 private school seniors and found that they applied to an average of 6.8 colleges with a standard deviation of 2.1. Do private school students apply to significantly more colleges than all high school seniors do? Set  = .05

40. Step One • We assume random sampling, level of measurement is interval-ratio, and that the sampling distribution is normal.

41. Step Two

42. Step Three • Sampling Distribution = t distribution •  = .05 • df = 24 • tcritical= 1.71

43. Step Four

44. Step Five • We reject the null hypothesis. We are 95% confident that private school students apply to significantly more colleges than seniors on the whole.

45. Probability of error & correct decision • The alpha level you set is your chance of Type I error (false rejection of H0) • There is also a probability of Type II error (false acceptance of H0) • Both the type of error possible and its probability depend upon which hypothesis is true (which you will never know). So you word carefully. • Example: If the null hypothesis were true that seniors from private schools applied to the same number of colleges, there is a 100% chance that you made a Type I error. But there is only a 5% (alpha) chance that you would make a Type I error if H0 were true.