• I. Next to each of the first five histograms, write a concise description of what each is a histogram of.
• II. Estimated population variance:
  (A) On each histogram of the sample variances (i.e., S_V_N and S_V_NMO), circle the distribution's mean that is printed in the figure legend.
  (B) On each histogram of the sample variances (i.e., S_V_N and S_V_NMO), mark its mean with a vertical line.
  (C) Circle and annotate the point in the SPSS program, printed at the very beginning of the output, that specifies the variance of the population distribution. (To see the commands in the output, be sure you have them displayed in the output. Edit -> Options -> Viewer tab -> In the "Initial Output State" box, select item Log and check the box at the bottom that says "Display commands in the log".)
  (D) Which form of sample variance, the form that divides by N or the form that divides by N-1, has expected value closest to the population variance? Write your answer next to the corresponding histogram.
• III. Critical values for z:
  (A) From the z-tables in the textbook, what are the critical z values for a 5% type-I error rate, one-tailed and two-tailed? Write these next to the sampling distributions of z (fourth and sixth histograms).
  (B) Consider the sampling distributions of z generated by SPSS. According to the percentiles listed in the table near the beginning of the SPSS printout, what are the critical z values for a 5% type-I error rate, one-tailed and two-tailed?
  (C) With vertical lines, mark the textbook critical values and the SPSS-generated critical values on the second z histogram; i.e., on the sixth, next-to-last histogram.
• IV. Critical values for t:
  (A) From the t-tables in the textbook, what are the critical t values for a 5% type-I error rate, one-tailed and two-tailed? What df do you use and why? Write your answer next to the sampling distributions of t (i.e., the fifth and seventh histograms).
  (B) Consider the sampling distributions of t generated by SPSS. According to the percentiles listed in the table near the beginning of the SPSS printout, what are the critical t values for a 5% type-I error rate, one-tailed and two-tailed?
  (C) With vertical lines, mark the textbook critical values and the SPSS-generated critical values on the second t histogram; i.e., on the seventh, last histogram.

• I. Critical value of t from null hypothesis:
  (A) What is the sample size used in this program? What are the df in this case?
  (B) From the Appendix in the textbook, report the critical t values for a one-tailed test with Type-I error rates of 10%, 5% or 1%. Make sure you use the correct df. (Note: I'm having you do this for a one-tailed test, despite the fact that you'll never use a one-tailed test in real research, only becaue the Monte Carlo results are more stable for larger percentages; that is, the Monte Carlo percentages will better match the theoretical values in the t-table.)
  (C) On the SPSS printout of percentiles, circle and annotate the corresponding estimated critical values for t.
  (D) On the second histogram (i.e., the graph with x-axis range -3 to 7), mark these correct and SPSS-estimated critical values, using vertical lines.
• II. Power for small effect size.
  (A) Circle and annotate the part of the SPSS program that specifies a population distribution with a small effect size (d=.2).
  (B) Using the SPSS output table, estimate the percentile, in this alternative hypothesis sampling distribution, of the 5%-critical value in the null hypothesis sampling distribution. What, therefore, is the power of this test?
  (C) Find the corresponding power listed in Table 7-9, p. 260, and report it.
  (D) On the fourth graph, mark the 5% critical value with a vertical line, and mark the area of the histogram that corresponds to the power of the test. (Notice that the t distribution is not symmetrical.)
• III. Power for moderate effect size.
  (A) Circle and annotate the part of the SPSS program that specifies a population distribution with a moderate effect size (d=.5).
  (B) Using the SPSS output table, estimate the percentile, in this alternative hypothesis sampling distribution, of the 5%-critical value in the null hypothesis sampling distribution. What, therefore, is the power of this test?
  (C) Find the corresponding power listed in Table 7-9, p. 260, and report it.
  (D) On the sixth graph, mark the 5% critical value with a vertical line, and mark the area of the histogram that corresponds to the power of the test. (Notice that the t distribution is not symmetrical.)
• IV. Power for large effect size.
  (A) Circle and annotate the part of the SPSS program that specifies a population distribution with a large effect size (d=.8).
  (B) Using the SPSS output table, estimate the percentile, in this alternative hypothesis sampling distribution, of the 5%-critical value in the null hypothesis sampling distribution. What, therefore, is the power of this test?
  (C) Find the corresponding power listed in Table 7-9, p. 260, and report it.
  (D) On the eighth graph, mark the 5% critical value with a vertical line, and mark the area of the histogram that corresponds to the power of the test. (Notice that the t distribution is not symmetrical.)

If you want more practice, try exercises #17 and #20

After	Before
5	9
4	7
7	8
6	5
9	6
8	3
11	4
10	1
12	2

• A. Do a one-sample t-test on the difference scores. Can we reject the null hypothesis that the self-help seminar has no effect on mean self-confidence?
• B. Do a paired-samples t-test on the after and before scores. Can we reject the null hypothesis that the self-help seminar has no effect on mean self-confidence?
• C. The two tests (one-sample and paired-sample) should agree exactly. Do they?
• D. Is there a significant relationship between the before and after scores? The answer is YES, despite the fact that there is not a significant difference in the means before and after. What is the relationship and where is this indicated in the SPSS t-test output? (Hint: Consider a scatterplot of the before and after scores. What happens to people who start off with low self-confidence? What happens to people who start off with high self-confidence? Would you recommend this seminar to someone who is already feeling pretty good?)

• A. What is M in this case? (The answer is 3.0. Mark and annotate this on the SPSS printout.)
• B. What is SE_M in this case? (The answer is 1.7078. Mark and annotate this on the SPSS printout.)
• C. What is t_crit in this case? Look it up (for 8 df, 5%, two-tailed).
• D. Verify that the CI in the SPSS output is M-t_critSE_M to M+t_critSE_M. That is, write down these formulas and show the equivalence with the SPSS confidence interval.