1. © 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 10. Hypothesis Testing II: Single-Sample Hypothesis Tests: Establishing the Representativeness of Samples

2. © 2008 McGraw-Hill Higher Education Single-Sample Hypothesis Tests Used to answer the question: “Is a parameter equal to some chosen target value?”

3. © 2008 McGraw-Hill Higher Education Sources of Target Values 1. A known population parameter from a comparison group 2. Known parameters from a past time period 3. A statistical ideal 4. A population is sampled, and the sample statistics are compared to known population parameters to determine whether the sample is representative of the population

4. © 2008 McGraw-Hill Higher Education Small Single-Sample Means Test Useful for testing a hypothesis that the mean of a population is equal to a target value

5. © 2008 McGraw-Hill Higher Education When to Use a Small Single Sample Means Test 1.There is only one variable 2. The level of measurement of the variable is interval/ratio 3. There is a single sample and population 4. n < 121 cases 5. There is a target value of the variable to which we may compare the mean of the sample

6. © 2008 McGraw-Hill Higher Education The Student’s t Sampling Distribution The sampling distribution curve used with especially small samples (n < 121 cases) is called the Student’s t-distribution The t-distribution is an approximately normal distribution

7. © 2008 McGraw-Hill Higher Education The t-distribution of Means For a sampling distribution of means, when the sample size is 121 or fewer cases, the probability curve begins to flatten into a t-distribution See Figure 10-1 in the text

8. © 2008 McGraw-Hill Higher Education Features of the t-distribution Standardized scores for the t-distribution are called t-scores and are computed like Z-scores The t-distribution, like the Z-distribution of the normal curve, allows us to calculate probabilities The t-distribution is symmetrical with mean, median, and mode centered, but the curve is flatter reflecting greater sampling error

9. © 2008 McGraw-Hill Higher Education The t-distribution Table The t-distribution table (Appendix B, Statistical Table C) is organized differently from the normal curve table See Table 10-1 in the text The t-distribution table provides t-scores for only the critical probabilities of.05,.01, and.001; that is, this table provides “critical t-scores”

10. © 2008 McGraw-Hill Higher Education Degrees of Freedom Degrees of freedom ( df ) are the number of opportunities in sampling to compensate for limitations, distortions, and potential weaknesses in statistical procedures Use of the t-distribution table requires the calculation of degrees of freedom For single-sample means tests, df = n - 1

11. © 2008 McGraw-Hill Higher Education More on Degrees of Freedom From repeated sampling we know that any single sample is only an estimate. An estimate can be distorted by limitations of the statistical procedures used to obtain it E.g., the mean is influenced by outliers The larger the sample, the greater the opportunity for an outlier to be neutralized in such a way as to not distort a sample mean A small sample is especially vulnerable to outliers

12. © 2008 McGraw-Hill Higher Education Z-Tests and t-Tests The small sample t-test applies when the sample size, n < 121; we use the t-distribution table However, regardless of sample size, all single-sample means tests are typically referred to as t-tests because the Z-test is a special case of a t-test, a t-test where n > 121; the t-distribution table includes critical scores for this situation under df = ∞

13. © 2008 McGraw-Hill Higher Education Features of a Single-Sample Means Test Follow the six steps of statistical inference Step 1: In general, state the H 0 : Parameter (sampled population) = known target parameter That is, the mean of a population = a target value As in all hypotheses tests, the H 0 and H A refer to population parameters

14. © 2008 McGraw-Hill Higher Education Features of a Single- Sample Means Test (cont.) Step 2. t-distribution with df = n - 1 The standard error is estimated by dividing the sample standard deviation by the square root of the sample size, n Draw the t-distribution curve Obtain the critical t-score from the t-table in Appendix B

15. © 2008 McGraw-Hill Higher Education The Critical Test Score for a t-Distribution Step 3: For a hypothesis test using the t- distribution curve, the critical test score is the specific t-score that is large enough to cause us to reject the H 0 After selecting the critical value from the t- distribution table, identify its location on the probability curve in Step 2 and estimate what kind of sample outcomes reach the critical region

16. © 2008 McGraw-Hill Higher Education Features of a Single-Sample Means Test (Continued) Step 4. The effect is the difference between the observed sample mean and the hypothesized target value The test statistic is the effect divided by the standard error The p-value is estimated from the t- distribution table

17. © 2008 McGraw-Hill Higher Education Relationships Among Mathematical Parts of the Test There are rules that assist in understanding the relationships among hypothesized parameters, observed sample statistics, the test statistic, the p- value, and the level of significance and its critical test score

18. © 2008 McGraw-Hill Higher Education Rule 1: The Test Effect and the Critical Test Score The H 0 is rejected when the test effect is large enough that the test statistic is greater than the critical test score value If │t observed │ > │ t α │, then p < α; reject H 0

19. © 2008 McGraw-Hill Higher Education Rule 2: The Effect and the Size of the p-Value The larger the test effect and the test statistic, the smaller the p-value A large test effect implies a significant difference between what was hypothesized and what was observed A large test effect makes the H 0 suspect

20. © 2008 McGraw-Hill Higher Education Rule 3: Rejection and Test Direction It is easier to reject with a one-tailed test than with a two-tailed test Choosing to use a one-tailed test must be done on the basis of theory or logic and is done in Step 1 of the six steps Do not look to Step 4, the sample outcome, to make a choice of test direction

21. © 2008 McGraw-Hill Higher Education Rule 4: The Level of Significance and Rejection The lower the level of significance, the harder it is to reject the H 0 The H 0 is rejected when the p-value is less than or equal to α. The p-value must be especially small to be smaller than a small α

22. © 2008 McGraw-Hill Higher Education Rule 5: Fail to Reject When Findings are in the Wrong Direction When using a one-tailed test but the observed sample statistic falls in the opposite direction, immediately fail to reject the H 0 Even if found statistically significant, one cannot justify a wrong direction finding

23. © 2008 McGraw-Hill Higher Education Testing for Sample Representativeness Single-sample hypothesis tests are especially useful for determining whether a sample is representative of the population from which it was drawn To establish the representativeness of a sample, data must be acquired on some known parameters of the population to provide the target value for the test

24. © 2008 McGraw-Hill Higher Education Testing for Sample Representativeness (cont.) Single-sample hypothesis tests determine whether the test effect is (a) due to random sampling error or (b) due to a failure to sample and represent some aspect of the population adequately

25. © 2008 McGraw-Hill Higher Education Testing for Sample Representativeness (cont.) The H 0 is stated as “the sample is representative” If the difference between the known parameter and the sample statistic is not statistically significant but simply due to sampling error, then we will not reject this H 0

26. © 2008 McGraw-Hill Higher Education Testing for Sample Representativeness (cont.) The H A is stated as “the sample is not representative” If the difference between the known parameter and the sample statistic is significantly different, then we reject the H 0 and conclude that the sampled population and targeted population are not one and the same – that there is sample bias

27. © 2008 McGraw-Hill Higher Education Testing for Sample Representativeness (cont.) Most tests of sample representativeness use nominal/ordinal variables, such as gender and race, because known parameters are available in government documents (e.g., the U.S. Census) These parameters provide the target values for such tests

28. © 2008 McGraw-Hill Higher Education Large Single-Sample Proportions Test The large single-sample proportions test is used for testing sample representativeness with nominal/ordinal variables

29. © 2008 McGraw-Hill Higher Education When to Use a Large Single- Sample Proportions Test 1.There is only one variable 2.It is a nominal/ordinal variable with P = p [of success category] and Q = p [of failure category/categories] 3. There is a single sample and population 4. The sample size is sufficiently large that (p smaller ) (n) > 5 5. There is a target value (known parameter) with which to compare the sample proportion

30. © 2008 McGraw-Hill Higher Education Features of a Large Single- Sample Proportions Test Step 1. Stating the H 0 : Parameter (sampled population) = a known target parameter That is: P u = a known target value

31. © 2008 McGraw-Hill Higher Education Features of a Large Single- Sample Proportions Test (cont.) Step 2. The sampling distribution of proportions is the approximately normal t- distribution with a standard error estimated by taking the square root of P u times Q u divided by n In the t-distribution table, look to df = ∞; this is essentially a normal curve test

32. © 2008 McGraw-Hill Higher Education Features of a Large Single- Sample Proportions Test (cont.) Step 4. The effect is the difference between the observed sample proportion (P s ) and the hypothesized target value (P u ) The test statistic is the effect divided by the standard error The p-value is estimated from the t- distribution table

33. © 2008 McGraw-Hill Higher Education Features of a Large Single- Sample Proportions Test (cont.) Steps 5 and 6: When the H 0 is rejected, focus on the interpretation of the H A, which is that the sample is NOT representative

34. © 2008 McGraw-Hill Higher Education Confidence Interval of the Mean When n is Small (n < 121 cases) For a confidence interval of the mean (Chapter 8), when the sample size is 121 or fewer cases, substitute the appropriate t-score in place of a Z-score t α in place of Z α

35. © 2008 McGraw-Hill Higher Education Statistical Follies: Sample Size and Tests of Representativeness When testing for sample representativeness, we wish to fail to reject the of H 0 of no significant difference With a small sample and its large standard error, it is too easy to accomplish this, but there is the chance of Type II error, in this case, concluding the sample is representative when it is not

36. © 2008 McGraw-Hill Higher Education Statistical Follies: Sample Size and Tests of Representativeness (cont.) To avoid a Type II error, the sample size must be sufficiently large Statistical power is a test statistic’s probability of not incurring a Type II error for a given level of confidence