One-sample T-Test of a Population Mean

# One-sample T-Test of a Population Mean

## One-sample T-Test of a Population Mean

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1. One-sample T-Test of a Population Mean Key Points about Statistical Test Sample Homework Problem Solving the Problem with SPSS Logic for One-sample T-Test of a Population Mean Power Analysis

2. One-sample T-Test: Purpose • Purpose: test whether or not the population mean represented by our sample has some specified value • Examples: • Social work students have higher GPA’s than other students • Social work students volunteer for more than 5 hours a week • UT social work students score higher on licensing exams than graduates of other programs • Social work students are getting younger every year

3. One-sample T-Test: Hypotheses • Hypotheses: • Null: population mean = specified value Versus • Research: population mean < specified value • Research: population mean ≠ specified value • Research: population mean > specified value • Decision: • Reject null hypothesis if pSPSS ≤ alpha (≠ relationship) • Reject null hypothesis if pSPSS÷2 ≤ alpha (< or > relationship)

4. One-sample T-Test: Assumptions and Requirements • Variable is interval level (ordinal with caution) • Variable is normally distributed • Acceptable degree of skewness and kurtosis or • Using the Central Limit Theorem

5. One-sample T-Test: Effect Size • Cohen’s d measures difference in means in standard deviation units. • Cohen’s d = population mean – specified value population standard deviation • Interpretation: • small: d = .20 to .50 • medium: d = .50 to .80 • large: d = .80 and higher

6. One-sample T-Test: APA Style • A one-sample T-test is presented as follows: • t(75) = 2.11, p = .02 (one –tailed), d = .48 Degrees of freedom Value of statistic Significance of statistic Include if test is one-tailed Effect size if available

7. Homework problems: One-sample t-test of a population mean This problem analyzes the variable "income" [rincom98] for a subset of the cases in GSS2000R.Sav. The subset is based on the variable "favor or oppose death penalty for murder" [cappun]. Using an alpha of .05, is the following statement true, true with caution, false, or an incorrect application of a statistic? Previous research on survey respondents who favored the death penalty for persons convicted of murder found that the average "income" was 12.30. Based on a one-sample t-test, the average "income" for survey respondents who favored the death penalty for persons convicted of murder (M = 13.41) is significantly different from the expected average based on previous research (mu = 12.30). • True • True with caution • False • Incorrect application of a statistic This is the general framework for the problems in the homework assignment on one-sample t-tests of population means. The description is similar to findings one might state in a research article.

8. Homework problems: Data set, variables, and sample This problem analyzes the variable "income" [rincom98] for a subset of the cases in GSS2000R.Sav. The subset is based on the variable "favor or oppose death penalty for murder" [cappun]. Using an alpha of .05, is the following statement true, true with caution, false, or an incorrect application of a statistic? Previous research on survey respondents who favored the death penalty for persons convicted of murder found that the average "income" was 12.30. Based on a one-sample t-test, the average "income" for survey respondents who favored the death penalty for persons convicted of murder (M = 13.41) is significantly different from the expected average based on previous research (mu = 12.30). • True • True with caution • False • Incorrect application of a statistic • The first two paragraphs identify: • The data set to use, e.g. GSS2000R.Sav • The subset of cases to include in the analysis • The variable to use to create the subset • The variable used in the t-test comparison of sample and population means • Thealpha level to use in the hypothesis test

9. Homework problems: One-sample t-test of a population mean This problem analyzes the variable "income" [rincom98] for a subset of the cases in GSS2000R.Sav. The subset is based on the variable "favor or oppose death penalty for murder" [cappun]. Using an alpha of .05, is the following statement true, true with caution, false, or an incorrect application of a statistic? Previous research on survey respondents who favored the death penalty for persons convicted of murder found that the average "income" was 12.30. Based on a one-sample t-test, the average "income" for survey respondents who favored the death penalty for persons convicted of murder (M = 13.41) is significantly different from the expected average based on previous research (mu = 12.30). • True • True with caution • False • Incorrect application of a statistic • The second paragraph identifies: • The value that we will use as the population mean • The third paragraph specifies: • The value of the sample mean for the subset of cases included in the problem • The relationship for deriving the research hypothesis

10. Homework problems: Choosing an answer The answer to a problem will be True if the t-test supports the finding in the problem statement. The answer to a problem will be True with caution if the t-test supports the finding in the problem statement, but the variable used is ordinal level. This problem analyzes the variable "income" [rincom98] for a subset of the cases in GSS2000R.Sav. The subset is based on the variable "favor or oppose death penalty for murder" [cappun]. Using an alpha of .05, is the following statement true, true with caution, false, or an incorrect application of a statistic? Previous research on survey respondents who favored the death penalty for persons convicted of murder found that the average "income" was 12.30. Based on a one-sample t-test, the average "income" for survey respondents who favored the death penalty for persons convicted of murder (M = 13.41) is significantly different from the expected average based on previous research (mu = 12.30). • True • True with caution • False • Incorrect application of a statistic • The answer to a problem will Incorrect application of a statistic if • the t-test violates the level of measurement requirement, i.e. the variable is nominal level • the assumption of normality is violated and the central limit theorem cannot be applied The answer to a problem will be False if the t-test does not support the finding in the problem statement.

11. Solving the problem with SPSS: Selecting the subset - 1 Our next task in SPSS is to select the subset cases that will be used in the analysis. The problem statement tell us “The subset is based on the variable "favor or oppose death penalty for murder" [cappun].” and that we are specifically interested in “survey respondents who favored the death penalty for persons convicted of murder…” Our first task is to find the data value for cappun which represents survey respondents who favored the death penalty for persons convicted of murder. We go to the Variable View in the SPSS Data Editor and locate the variable.

12. Solving the problem with SPSS: Selecting the subset - 2 We scroll to the right until we see the Values column. When we click on the cell for cappun in the values column, a button with an ellipsis on it appears. Click on this button to open the Values Label dialog box. Click on OK to close the dialog box. The Values Labels dialog box shows us the text labels that the creator of the data set assigned to each of the possible numeric responses for this variable. 1 = “FAVOR” would be the logical choice to indicate respondents who favored the death penalty . This analysis will include cases who have a score of 1 for the variable cappun.

13. Solving the problem with SPSS: Selecting the subset - 3 To select the subset of cases for this analysis, we return to the Data View of the SPSS Data Editor and we choose the Select Cases… command from the Data menu.

14. Solving the problem with SPSS: Selecting the subset - 4 In the Select Cases dialog box, we mark the option button If condition is satisfied, and click on the If… button which becomes active when the option button is marked.

15. Solving the problem with SPSS: Selecting the subset - 5 Second, we click on the right arrow button to move the variable to the text box where we will compose our selection criteria. First, we highlight the variable we want to use, cappun, in selecting the subset.

16. Solving the problem with SPSS: Selecting the subset - 6 First, we complete the selection criteria by typing the value for the cases we want to include, = 1. Second, we click on the Continue button to close the Select Cases: If dialog box.

17. Solving the problem with SPSS: Selecting the subset - 7 When we return to the Select Cases dialog, we see that SPSS has printed our selection criteria next to the If… button. Click on the OK button to complete the selection of the subset.

18. Solving the problem with SPSS: Selecting the subset - 8 When we return to the Data Editor, we scroll the variables to the right until we see the column for cappun. We see that SPSS has marked out the cases that will be excluded by drawing a diagonal slash through the row number. The cases that are excluded have either a “2” for “OPPOSE” answers or an “8” or “9” which indicate missing data specified by the creator of the data set. The cases with a value of “1” for cappun do not have the slash and will be included in the analysis.

19. Solving the problem with SPSS:Level of measurement Statistical tests of means require that the variable be interval level. "Income" [rincom98] is an ordinal level variable, which violates this requirement in the strictest interpretation of level of measurement. However, since the research literature often computes means for ordinal level data, especially scaled measures, we will follow the convention of applying interval level statistics to ordinal data. Since all analysts may not agree with this convention, a caution is added to any true findings.

20. Solving the problem with SPSS: Evaluating normality - 1 The one-sample t-test uses the t-distribution for the probability of the test statistic. To obtain accurate probabilities, the variable must follow a normal distribution. We will generate descriptive statistics to evaluate normality. Select the Descriptive Statistics > Descriptives… command from the Analysis menu.

21. Solving the problem with SPSS: Evaluating normality - 2 First, move the variable we will use in the t-test, rincom98, to the Variable(s) list box. Second, click on the Options… button to select the statistics we want.

22. Solving the problem with SPSS: Evaluating normality - 3 First, in addition to the statistics, SPSS has checked by default, mark the Kurtosis and Skewness check boxes on the Distribution panel. Second, click on the Continue button to close the dialog box.

23. Solving the problem with SPSS: Evaluating normality - 4 Click on the OK button to obtain the output.

24. Solving the problem with SPSS: Evaluating normality - 5 "Income" [rincom98] satisfied the criteria for a normal distribution. The skewness of the distribution (-.723) was between -1.0 and +1.0 and the kurtosis of the distribution (-.234) was between -1.0 and +1.0.

25. For this problem, we have a sample size, N, of 111, clearly large enough to apply the central limit theorem if needed. Solving the problem with SPSS: Evaluating normality - 6 If we were unable to establish normality by examining skewness and kurtosis, it is still possible to satisfy the assumption by applying the central limit theorem. The central limit theorem states that the distribution of sample means will be normal even it the variable is not, provided the sample size is 30 or more. This implies that the probabilities we obtain for making a decision about the hypotheses are accurate. If we are unable to establish normality either by the distribution or by the central limit theorem, we should not use the t-test.

26. Solving the problem with SPSS: The one-sample t-test - 1 Having satisfied the level of measurement requirement for the t-test, and the assumption of normality, we now do the actual t-test. Select Compare Means > One-Sample T Test… from the Analyze menu.

27. Solving the problem with SPSS: The one-sample t-test - 2 The finding we are trying to verify is: Based on a one-sample t-test, the average "income" for survey respondents who favored the death penalty for persons convicted of murder (M = 13.41) is significantly different from the expected average based on previous research (mu = 12.30). First, move the variable rincom98 to the Test Variable(s) list box. Third, click on the OK button to generate the output. Second, enter the population mean, mu, into the Test Value text box.

28. Solving the problem with SPSS: Answering the question - 1 The finding we are trying to verify is: Based on a one-sample t-test, the average "income" for survey respondents who favored the death penalty for persons convicted of murder (M = 13.41) is significantly different from the expected average based on previous research (mu = 12.30). Our first task is to make certain we have solved the right problem. First, we check to make certain we have entered the population mean correctly as the Test Value. Second, we verify that the mean of our sample, as computed by SPSS, matches the value of the sample mean stated in the problem.

29. Solving the problem with SPSS: Answering the question - 2 Since the problem states that the mean for our data is significantly different from the mean found in previous research, we test the two-tailed hypothesis that mu is not equal to12.30 versus the null hypothesis that mu is equal to 12.30. The one-sample t-test for this problem produced the statistical result: t(110) = 2.180, p = .03.

30. Solving the problem with SPSS: Answering the question - 3 Since the two-tailed probability for the t statistic is less than or equal to the alpha level of 0.05 as stated in the problem, we reject the null hypothesis and support the research hypothesis that there is a significant difference. The answer to the question is True with caution. The caution is added because the variable rincom98 was ordinal level. Our sample is either unlikely to be from the same population reported in previous research, or some event has altered the population mean.

31. Solving the problem with SPSS: Probability for a one-tailed t-test If the problem had stated that the mean for our data is significantly higher than the mean found in previous research, we would have tested the one-tailed hypothesis that mu is greater than 12.30 versus the null hypothesis that mu is equal to 12.30. SPSS only produces the two-tailed probability, but we can easily obtain the one-tailed probability by dividing the two-tailed probability by 2. If this problem has stated a one-tailed research hypothesis, we would have compared 0.015 (.031 ÷ 2) to the alpha level of .05 to make our decision about the null hypothesis.

32. Restoring all of the cases to the dataset - 1 We have selected a specific subset of cases for this problem. To make sure we do not use the wrong subset for the next problem, we will restore all of the cases to the data set. Click on the Select Cases… command from the Data menu.

33. Restoring all of the cases to the dataset - 2 Click on the All cases option button to remove the If condition. Click on the OK button to complete the command.

34. Restoring all of the cases to the dataset - 3 The slashes through the case numbers are removed, indicating that all of the cases are available to the next command.

35. Logic for one-sample t-test:Level of measurement Select subset of cases specified in problem Measurement level of variable? Interval/ordinal Nominal (dichotomous) Inappropriate application of a statistic

36. Logic for one-sample t-test:Assumption of normality Sample size is at least 30? Skewness and Kurtosis between -1.0 and +1.0? No No Inappropriate application of a statistic Yes Yes

37. Logic for one-sample t-test:Decision about null hypothesis One-tailed or two-tailed test? Two-tailed One-tailed Divide two-tailed significance by 2 Probability for t-test less than or equal to alpha? Yes No Add caution for ordinal variable. True False

38. Power Analysis:One-sample T-test Problem that was False This problem analyzes the variable "occupational prestige score" [prestg80] for a subset of the cases in GSS2000R.Sav. The subset is based on the variable "favor or oppose death penalty for murder" [cappun]. Using an alpha of .01, is the following statement true, true with caution, false, or an incorrect application of a statistic? Previous research on survey respondents who opposed the death penalty for persons convicted of murder found that the average "occupational prestige score" was 41.52. Based on a one-sample t-test, the average "occupational prestige score" for survey respondents who opposed the death penalty for persons convicted of murder (M = 45.71) is significantly higher than the expected average based on previous research (mu = 41.52). 1 True 2 True with caution 3 False 4 Incorrect application of a statistic The answer to this problem was false because the probability for the t-test was 0.02, greater than the alpha of 0.01. We can conduct a post-hoc power analysis to determine if the number of available cases was sufficient to find a statistically significant difference.

39. Power Analysis:Statistical Results for One-sample T-test - 1 The answer to the problem was false because the one-tailed significance was p = .02 (.04 ÷ 2), greater than the alpha of .01.

40. Power Analysis:Statistical Results for One-sample T-test - 2 We can calculate the effect size for the data for this problem, Cohen’s d, by dividing the Mean Difference by the Std. Deviation.

41. Power Analysis:Statistical Results for One-sample T-test - 3 Using Microsoft Excel for the calculations, we find that the effect size is 0.27. Using Cohen’s interpretative guidelines, an effect size of 0.27 would be characterized as “small”. The power analysis question in this case will enable us to answer the question of whether or not the number of cases available for this analysis was sufficient to detect an effect size that was this small.

42. Access to SPSS’s SamplePower Program The UT license for SPSS does not include SamplePower, the SPSS program for power analysis. However, the program is available on the UT timesharing server. Information about access this program is available at this site.

43. Power Analysis for One-sample T-test - 1 In the SamplePower program on the ITS Timesharing Systems, select the New… command from the File menu.

44. Power Analysis for One-sample T-test - 2 First, select the Means tab to access the tests for means. Second, since the one-sample t-test evaluated our data against a specific value for the population mean, select the option button for One sample t-test that mean = specific value. Third, click on the Ok button to enter the specific values for our problem.

45. Power Analysis for One-sample T-test - 3 SPSS sets the default test to a two-tailed test with an alpha of .05. Since our test was a one-tailed test with an alpha of .01, we click on the text specified as the SPSS default.

46. Power Analysis for One-sample T-test - 4 Second, click on the 1 Tailed option on the Tails panel. First, click of 0.01 option button on the Alpha panel. Third, click on the Ok button to change the test specifications.

47. Power Analysis for One-sample T-test - 5 • We enter the values from the SPSS output from the one-sample t-test: • 45.71 for Expected mean • 41.52 for Test against the constant • 15.34 for Standard Deviation • 59 for the N of Cases When we enter the values, SPSS computes the power of our test on the slider bar. Power of 0.39 is well below Cohen’s recommendation of 0.80. At, 0.39, we have less than a 50-50 chance of finding statistical significance for a small effect.

48. Power Analysis for One-sample T-test - 6 To find the sample size we would have needed to achieve power of 0.80, we can click on the up-arrow of the spin box, and watch the power value in the slider bar. Increasing the sample size to 90 would have increased our power to 0.59.

49. Power Analysis for One-sample T-test - 7 We would have needed a sample of about 140 to reach the target of power = 80%.

50. Power Analysis for One-sample T-test - 8 To find the exact sample size needed, select Find N for power of 80% from the Tools menu. The power analysis question in this case will enable us to answer the question of whether or not the number of cases available for this analysis was sufficient to detect an effect size that was this small.