1 Million+ Step-by-step solutions

Calculate both LSD and HSD for the two experiments in Exercise 3 of the previous chapter, using the harmonic mean of the three sample sizes as your value for n. What conclusions can you draw for each experiment? Use your values for LSD and HSD to compare the relative statistical power of these two procedures.

**Data from Exercise 3(previous chapter)**

For each of the following two experiments, calculate the means and variances for each group first, and then use those statistics to perform the ANOVA.

a. Can you reject the null hypothesis for Experiment 1? Display your results in an ANOVA summary table, and state the significance of your F ratio in a sentence, using the proper format. Include the value for eta squared.

b. Can you reject the null hypothesis for Experiment 2? Display your results in an ANOVA summary table, and state the significance of your F ratio in a sentence, using the proper format. Include an estimate of omega squared.

Consider the following hypothetical results, as published in a hypothetical journal: “The participants had been divided equally among three experimental conditions negative feedback, positive feedback, and no feedback —and the mean scores for the three groups at the end of the study were 11.1, 14.7, and 12.4, respectively. Moreover, the group differences were found to be statistically significant, F (2, 57) = 3.69, p < .05.”

(a) Find the value of η^{2} for these data. How would you describe the size of this effect?

(b) Estimate ω^{2} for the population represented by these groups.

The following data table comes from Exercise 4 in Chapter 8, in which a psychology professor used three different methods of instruction in three small classes. In Chapter 8 you were asked to compare each pair of groups with the nonparametric rank-sum test. This time we would like you to:

(a) Test the null hypothesis that all three methods of instruction have the same effect on examination scores, using the Kruskal-Wallis H test, and compute Î·^{2}/R;

(b) repeat the analysis by computing the ordinary one-way ANOVA and Î·2;

(c) compare the results of parts a and b.

For each of the two (separate) experiments that follow, perform the Kruskal-Wallis H test. If an experiment yields statistically significant results, compute the appropriate eta squared.

Calculate the one-way ANOVA and determine its statistical significance for the following data:

For each of the following two experiments, calculate the means and variances for each group first, and then use those statistics to perform the ANOVA.

a. Can you reject the null hypothesis for Experiment 1? Display your results in an ANOVA summary table, and state the significance of your F ratio in a sentence, using the proper format. Include the value for eta squared.

b. Can you reject the null hypothesis for Experiment 2? Display your results in an ANOVA summary table, and state the significance of your F ratio in a sentence, using the proper format. Include an estimate of omega squared.

Following are two separate (hypothetical) sets of data that are somewhat exaggerated to help clarify the procedures underlying analysis of variance. In each case, the experimenter is interested in conditions that affect the number of errors made by participants on a simple clerical task that must be performed quickly. Group 1 listens to upbeat dance music while performing the tasks; Group 2 listens to soothing, New Age music; and Group 3 listens to white noise (e.g., the steady hum of a machine in the background).

a. By inspection, in which case would you guess that the difference among groups is more likely to be statistically significant? Why?

b. Carry out the analysis of variance for Experiment 1. Are the results significant at the .05 level? Calculate eta squared.

c. Carry out the analysis of variance for Experiment 2. Are these results significant at the .05 level? Calculate eta squared.

d. Briefly describe why the F ratio and eta squared statistics differ dramatically for the two experiments even though the sample means (i.e., 2, 6, and 4) are the same in both cases.

e. Calculate an unbiased estimate of Ï‰^{2} for whichever experiment is significant.

a. Calculate the power your test would have in Exercise 7 of Chapter 7 if the two sets of scores were not correlated at all in the population (use g for those data as d in your power calculation).

b. Calculate the power your test would have in Exercise 7 of Chapter 7 if the two sets of scores had a population correlation of ρ = .5 (again, use g for those data as d in your power calculation).

Calculate the missing values in the following table, assuming that you are comparing two independent, equalsize samples with a two-tailed t test. (Note: Small = .2, Medium = .5, Large = .8.)

a. Calculate g from the pooled-variance t value you computed to solve Exercise 2 in Chapter 7. Use that value for g as your estimate of d, and compute the sample sizes you would need for power = .9, for a .05 two tailed test, assuming that you plan to use two equal-size samples to compare Turck and Dupre Halls.

b. Given the sizes of the samples in Exercise 2 of Chapter 7, how large would d have to be to obtain power = .8, for a .05, two-tailed test?

a. Calculate g for the comparison of the means of Turck and Kirk Halls (if you did not already calculate that value for Exercise 1 of Chapter 7). Use g as your estimate of d, and compute the sample sizes you would need for power = .75, for a .05 two-tailed test, comparing the two dormitories.

b. Given the sizes of the samples from Turck and Kirk Halls, how large would d have to be to obtain power = .85, for a .01, two-tailed test?

a. A personality theorist expects that two particular traits have a linear correlation on the order of .40 in the population she plans to sample. She wishes to test the null hypothesis that the correlation between the two traits is exactly zero, using the .05 criterion of significance and a random sample of 65 subjects. Is the power of this statistical test satisfactory? Explain.

b. How large a sample would the theorist in part (a) need to obtain power = .9 with alpha set at .01 for a twotailed test? For a one-tailed test?

a. A politician needs 50% or more of the vote to win an election. To find out how his campaign is going, he plans to obtain a random sample of 81 voters and see how many plan to vote for him. He is willing to posit a specific alternative hypothesis of 60% (or 40%) and wishes to use the .01 criterion of significance. Compute the power of the statistical test. How would you evaluate this research plan?

b. Suppose the politician decides to switch to the .05 criterion of significance (but that the other values are not changed). Will this improve the power of the statistical test to a satisfactory level?

c. The politician finally resigns himself to doing more work and obtaining a larger sample. He wants power to be .75. How large a sample does he need (using the .05 criterion)?

Suppose that the students at Bigbrain University are planning to test whether the mean math SAT score for their school is higher than the national average (μ) of 500 (assume that σ = 100). a. If they believe that their mean is 520, and they plan to sample 25 students, what is the power of their statistical test at the .05 level, two-tailed? What would the power be for a one-tailed test? Explain why power is higher for the one tailed test.

a. If they believe that their mean is 520, and they plan to sample 25 students, what is the power of their statistical test at the .05 level, two-tailed? What would the power be for a one-tailed test? Explain why power is higher for the one-tailed test.

b. Recalculate the one- and two-tailed power values in part (a) assuming that the Bigbrain students expect their average to be 50 points higher than the national average. Explain why these power values are higher than the ones you calculated in part (a).

c. Redo part (a) assuming that a sample of 100 Bigbrain students is being planned. Explain why these power values are higher than the ones you calculated in part (a).

d. Given the expected effect size in part (a), what sample size would be needed to obtain a power of .8 for a two-tailed .05 test? For a two-tailed .01 test?

e. Repeat part (d) given the expected effect size in part (b).

Another industrial psychologist asks a group of 9 assembly-line workers and 11 workers not on an assembly line (but doing similar work) to indicate how much they like their jobs on a 9-point scale (9 = like, 1 = dislike). The results are as follows:

(a) Test the null hypothesis that there is no relationship between the assembly line variable and job satisfaction by computing r_{pb} and then using Table C to find the critical value. What should the psychologist decide?

(b) Calculate the corresponding t value using Formula 10.7, and then compute an estimate of omega squared (Ï‰^{2}) by using Formula 10.10. Does it look like there is a fairly large effect of job type on job satisfaction in the population?

An industrial psychologist obtains scores on a job-selection test from 41 men and 31 women, with the following results (see Exercise 4 in Chapter 7): men, M = 48.75 (SD = 9.0); women, M = 46.07 (SD = 10.0). First, calculate g for these data. Then use the appropriate formula to calculate r_{pb} directly from g. What proportion of variance in these data is accounted for by gender?

Use the t value you calculated for Exercise 1 in Chapter 7 to find the point-biserial r that corresponds to the difference in means between Turck and Kirk Halls. Then use the appropriate t formula to test the r_{pb }you just computed for significance. How does this t value compare with the original t value you calculated for Exercise 1 in Chapter 7?

Compute the linear regression equation for predicting Y from X for each of the data sets in Exercise 3 in Chapter 9. What proportion of the variance in Y is accounted for by X in each data set? For each data set, use the appropriate formula to calculate the unbiased standard error of the estimate (s_{est}).

**Data from exercise 3(chapter 9)**

Ten subjects participate in a problem-solving experiment. Two judges are asked to rank order the solutions with regard to their creativity (1 = most creative, 10 = least creative). The experimenter wishes to know if the judges are in substantial agreement. Following are the rankings; what should the experimenter decide?

The data from Exercises 8 and 9 in Chapter 7 are reproduced in the tables that follow.

a. Compute the Pearson r for Data Set 3, Chapter 1, and test for significance at the .01 level.

b. Use the z scores you calculated for these data in the exercises of Chapter 4 to recompute the r with the zproduct formula. Is it the same?

c. Compute the Pearson r for Data Set 3, Chapter 1, with X rearranged, and test for significance at the .05 level.

The data from Exercise 7 in Chapter 7 are reproduced in the following table. Calculate the mean and the unbiased standard deviations for both the experimental and control groups, and then compute the Pearson r with the raw-score formula that uses the s's rather than the Ïƒ's in its denominator. Is the correlation coefficient significant? Given your decision with respect to the null hypothesis, what type of error could you be making, Type I or Type II? (Advanced exercise: Recompute the matched-pairs t test with the formula that is based on Pearson's r, and compare it to the t value you calculated for these data in the previous chapter.)

**Data from exercise 7(chapter 7)**

Test the results for statistical significance at the .05 level. What should the psychologist decide about his or her new programmed text?

A college dean would like to know how well he can predict sophomore grade point average for first semester freshmen so that students who are headed for trouble can be given appropriate counseling. After students have been in school for one semester, the dean obtains their numerical final examination average for the first semester (based on a total of 100 points) and the average number of â€œcutsâ€ per class during the semester. He then waits for a year and a half, and when the students have finished their second year, he obtains their sophomore grade point average. To keep the computations down to a reasonable level, we will assume that the dean has a sample of only seven cases. (Note that in a real study of this kind there would be many more participants, but the same procedures would be used.)

a. Convert the test scores (X) and sophomore averages (Y) to z scores. By inspection of the paired z scores, estimate whether the correlation between these two variables is strong and positive, about zero, or strong and negative. Then verify your estimate by computing r using the z score product formula.

b. Use (raw-score) Formula 9.2 to compute the Pearson r between the number of cuts (C) and the sophomore average (Y). Can you reject the null hypothesis at the .05 level with a two-tailed test? (Use Table C.)

c. Repeat part (b) for the correlation between the number of cuts (C) and the test score (X). Is this correlation significant at the .01 level with a one-tailed test?

**Formula 9.2**

**Table C**

Ten subjects participate in a problem-solving experiment. Two judges are asked to rate the solutions with regard to their creativity (1 = most creative, 10 = least creative). The experimenter wishes to know if the judges were equally lenient in their ratings. Given the ratings in the table, what should the experimenter decide?

For each of the two (separate) experiments that follow, perform the Wilcoxon test. If an experiment yields statistically significant results, also compute r_{c}.

A psychology professor uses three different methods of instruction in three small classes, with the assignment of students to classes being random, and gives each class the same final examination. The results are in the table that follows.

(a) Perform the appropriate test to compare each possible pair of groups and, in each case, state which group had the higher scores and whether the results were statistically significant at the .01 as well as the .05 level.

(b) Calculate the appropriate measure of the strength of relationship between method of instruction and examination scores for each pair of groups, and comment briefly on the apparent size of the effect.

In Experiment 1 (see the following table), Ihno is comparing scores on a practice final exam between the students who performed most poorly on the midterm (Group 1) and those who got the highest midterm scores (Group 2). Experiment 2 also compares scores on the practice final, but in this case, Group 1 consists of the students who received the highest scores on the first quiz and Group 2 contains the students who received the lowest scores.

(a) Perform the rank-sum test for each experiment. Can you reject the null hypothesis in each case?

(b) Calculate the appropriate measure for the strength of relationship between the two variables, for each experiment. How strong does the relationship appear to be in each case?

Redo Exercise 5 from Chapter 7 using the rank-sum test instead of a t test. Does your statistical conclusion differ from the one you made in the previous chapter? Why would the rank-sum test not be very accurate for the data in that exercise?

**Data from exercise 5**

Suppose that the industrial psychologist from the previous exercise is testing the difference in performance on the job-selection test of two different ethnic groups. Given the following data, can the psychologist reject the null hypothesis (alpha = .05) that the population means of these two groups are the same for the job-selection test she is investigating? If these results are statistically significant, compute the 95% CI for the difference of the population means.

An operator of a certain machine must turn it off quickly if a danger signal occurs. To test the relative effectiveness of two types of signals, a small group of operators is randomly divided into two groups. Those in Group 1 operate machines with a newly designed signal, while those in Group 2 use machines with the standard signal. The signal is flashed unexpectedly, and the reaction time of each operator (time taken to turn off the machine after the signal occurs) is measured in seconds. The actual results (not ranks) are shown here.

(a) Use the rank-sum test to determine whether there is a significant difference between the two groups.

(b) Compute the appropriate measure of strength of relationship.

Suppose that you have read the following sentence in a psychology journal: “The experimental group obtained a higher score on the recall task (M = 14.7) than did the control group (M = 11.1), and this difference was statistically significant, t(38) = 2.3, p < .05.” Assuming that the two samples are the same size:

(a) Find the 95% CI for the difference of the means.

(b) Find g, the size of the effect in the sample data.

Now carry out a matched t test for the same set of data as in Exercise 8, but with the X values rearranged as shown.

For the following set of data, assume that the X score represents the participant's performance in the experimental condition and that the Y score represents the same person's performance in the control condition. (Thus, each participant serves as his or her own control.) Compute the matched t test for these data. Also compute the 95% CI for the mean of the difference scores.

An educational psychologist has developed a new textbook based on programmed instruction techniques and wishes to know if it is superior to the conventional kind of textbook. He therefore obtains participants who have had no prior exposure to the material and forms two groups: an experimental group, which learns via the programmed text, and a control group, which learns via the old-fashioned textbook. The psychologist is afraid, however, that differences among participants in overall intelligence will lead to large error terms. Therefore, he matches his participants on intelligence and forms 10 pairs such that each pair is made up of two people roughly equal in intelligence test scores. After both groups have learned the material, the psychologist measures the amount of learning by means of a 10-item quiz. The results are as follows:

Test the results for statistical significance at the .05 level. What should the psychologist decide about his or her new programmed text?

Repeat Exercise 5 for the following two sets of summary statistics. In each case, determine if the results would be significant for a one-tailed as well as a two-tailed test, but compute the CI only if the two-tailed test is significant. Regardless of statistical significance, compute g, the standardized measure of effect size in sample data, for each data set.

**Data from exercise 5**

Suppose that the industrial psychologist from the previous exercise is testing the difference in performance on the job-selection test of two different ethnic groups. Given the following data, can the psychologist reject the null hypothesis (alpha = .05) that the population means of these two groups are the same for the job-selection test she is investigating? If these results are statistically significant, compute the 95% CI for the difference of the population means.

An industrial psychologist obtains scores on a job-selection test from 41 men and 31 women, with the following results: men, M = 48.75(SD = 9.0); women, M = 46.07(SD = 10.0). Test this difference for significance at both the .05 and .01 levels (two-tailed). Determine the effect size for the difference of these two sample means. Does this effect size seem to be large or small?

Repeat Exercise 2, but this time compare Kirk Hall to Dupre Hall, and conduct only a one-tailed test at the .05 level, assuming that Kirk will have the larger mean.

**Data from exercise 2**

Repeat the previous exercise, except that this time compare Turck Hall to Dupre Hall. Would the t value based on the separate-variances formula (7.8) be smaller or larger than the pooled-variances t value?

**Data from previous exercise**

Use the two-sample t test to determine whether the difference in means between Turck and Kirk Halls is significant at the .05 and .01 levels, two-tailed. Given your decision with respect to the null hypothesis, which type of error could you be making: Type I or Type II? Report the results of your t test in a sentence that includes the means of the two dormitories and a measure of the effect size. Compute both the 95% and 99% confidence intervals for the difference of the two population means. Explain how the presence or absence of zero in each of those CIs is consistent with your decision regarding the null hypothesis at the .05 and .01 levels.

Repeat the previous exercise, except that this time compare Turck Hall to Dupre Hall. Would the t value based on the separate-variances formula (7.8) be smaller or larger than the pooled-variances t value?

**Data from previous exercise**

Use the two-sample t test to determine whether the difference in means between Turck and Kirk Halls is significant at the .05 and .01 levels, two-tailed. Given your decision with respect to the null hypothesis, which type of error could you be making: Type I or Type II? Report the results of your t test in a sentence that includes the means of the two dormitories and a measure of the effect size. Compute both the 95% and 99% confidence intervals for the difference of the two population means. Explain how the presence or absence of zero in each of those CIs is consistent with your decision regarding the null hypothesis at the .05 and .01 levels.

A certain business concern needs to obtain at least 20% of the market in order to make a profit. A random sample of 200 prospective buyers is asked whether they will purchase the product. What should the company conclude if (find the 95% CI in each case, in addition to performing the null hypothesis test):

a. 26 of those asked said they would buy the product?

b. 46 of those asked said they would buy the product?

c. 58 of those asked said they would buy the product?

A politician has staked his political career on whether a new state constitution will pass. To find out which way the wind is blowing, he obtains a random sample of 100 voters a few weeks prior to the election and finds that 60% of the sample says that they will vote for the new constitution. Assuming that the constitution will fail if it receives 50% or less of the vote, should he conclude that the electorate as a whole will support the new constitution? Perform the appropriate statistical test to answer this question.

This exercise is based on the following data set: 68.36, 15.31, 77.42, 84.00, 76.59, 68.43, 72.41, 83.05, 91.07, 80.62, 77.83.

a. Perform a t test in order to decide whether you can reject the null hypothesis of μ = 52.3 at the .01 level (two-tailed) for these data.

b. Redo your t test in part (a) for a null hypothesis of μ = 85.0.

c. Compute the 99% CI for the population mean from which these data were drawn. Explain how this CI could be used to draw conclusions about the null hypotheses in parts (a) and (b).

This exercise is based on the following data set: 1, 3, 6, 0, 1, 1, 2, 1, 4.

a. Perform a t test in order to decide whether you can reject the null hypothesis of μ = 2.5 at the .05 level (two tailed) for these data.

b. Redo your t test in part (a) for a null hypothesis of μ = 6.0.

c. Compute the 95% CI for the population mean from which these data were drawn. Explain how this CI could be used to draw conclusions about the null hypotheses in parts (a) and (b).

For Kirk Hall, compute each of the following:

a. The 99% CI.

b. The 95% CI.

c. The 90% CI.

d. Explain why the size of the CIs becomes larger as the level of confidence increases.

Answer these problems by calculating a t value and comparing it to the critical value for a two-tailed test at the .05 level. (Note: You can save time by using the means and standard deviations you calculated for the dormitories called Turck and Kirk Halls for previous exercises.)

a. Would you retain or reject the null hypothesis that the population mean for Turck Hall is 16?

b. Would you retain or reject the null hypothesis that the population mean for Kirk Hall is 11?

Calculate the mean and (unbiased) SD for the following sample: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95. Assume that the SD of the population from which these data were sampled is the same as the SD that you just calculated for the sample.

(a) Are these data consistent with the hypothesis that the mean of the population is 60? Explain.

(b) Repeat part (a) for a hypothesized population mean of 36.

This exercise is designed to give you a more direct understanding of the standard error of the mean. Create a population as follows: Get 20 identical small slips of paper or file cards. On eight of these slips, write the number 50; on five of the slips, write the number 51; on another five, write the number 49. For the last two slips of paper, write the number 48 on one and 52 on the other. Place all the slips in a bowl and mix thoroughly.

a. Draw one slip at random from the bowl and write down its number. Then replace the slip in the bowl, mix thoroughly, and draw at random again (this is called sampling with replacement). Keep repeating this process until you have written down the numbers for five random selections. Calculate the average for these five numbers. This is your first sample mean.

b. Repeat the process described in part (a) until you have calculated a total of six sample means.

c. Calculate the mean and (unbiased) SD of the six sample means you found in part (b). The latter statistic is the standard error of the mean calculated directly from the sample means rather than estimated from a sample SD divided by N. Is the mean of the sample means about what you expected?

d. Estimate the standard error of the mean separately from the SD of each of the six samples you drew. How do these six estimates compare to each other, and to the standard error you calculated directly in part (c)?

Repeat Exercise 4, parts (a) and (b), for the sample from Dupre Hall. What is the same and what is different between your calculations for this exercise, and your calculations for Exercise 4?

**Data from exercise 4**

a. Based on the data in Table 1.1, what is the z score for friendliness for the students sampled from Turck Hall? What are the one- and two-tailed p values for this sample?

b. If you were testing the null hypothesis (i.e., μ = 10), would you reject H_{0} for the students of Turck Hall at the .05 level for a one-tailed test? For a two-tailed test?

Referring to the 20-point friendliness measure used in the first exercise of Chapter 1, assume that the mean score for all American college students is known to be 10, with a SD of 5 points.

a. Based on the data in Table 1.1, what is the z score for friendliness for the students sampled from Turck Hall? What are the one- and two-tailed p values for this sample?

b. If you were testing the null hypothesis (i.e., Î¼ = 10), would you reject H_{0} for the students of Turck Hall at the .05 level for a one-tailed test? For a two-tailed test?

**Table 1.1**

Given that for SAT scores, μ = 500 and σ = 100:

a. Test the claim of students at Bigbrain University that they have SAT scores that are statistically significantly higher than the ordinary population, because a random sample of 25 of their students averaged 530 on this test. Make your statistical decision by comparing the z score for the Bigbrain sample with the appropriate critical value for a two-tailed test. Would a one-tailed test be significant in this case? Would it be justified?

b. Repeat part (a) for a random sample of 64 students, who also have a mean of 530 on the SAT.

Assume that the mean IQ for all 10th graders at a large high school (i.e., population) is 100 with σ = 15 and that the students are assigned at random to classes with an N of 25.

a. What is the z score for a class whose IQ averages 104? What is the one-tailed p value for this z score? What is the two-tailed p?

b. Repeat part (a) for a class whose mean IQ is 92.

c. Perform two-tailed null hypothesis tests for the classes in part (a) and part (b) at both the .05 and .01 levels.

In each case explain whether you could be making a Type I or Type II error.

Assume that the mean height for women at a large American university (to be viewed as a population) is 65 inches with a standard deviation of 3 inches.

a. If the women are placed randomly into physical education classes of 36 each, what will be the SD of the class means for height (i.e., the standard error of the mean)?

b. Using your answer to part (a), what is the z score for a phys ed class whose average height is 64.2 inches? What are the one- and two-tailed p values for this class?

c. If you were testing the null hypothesis (i.e., μ = 65), would you reject H_{0} for the class in part b, at the .05 level for a one-tailed test? For a two-tailed test?

d. Repeat part (b) for a class whose mean height is 67.4 inches. Would you reject the null hypothesis for this class with a two-tailed test at the .05 level? At the .01 level?

Between which two SAT scores would you find the middle:

(a) 70% of the normal distribution?

(b) 90% of the normal distribution?

A single student is drawn at random from the population. What is the probability that this student has a score:

a. Of 410 or less?

b. Between 430 and 530?

c. Between 275 and 375?

d. In the top 5% of the population?

Base your answers to this exercise on the percentages you found for the previous exercises.

A psychologist wishes to test a new learning strategy on the bottom 15% of those who took the math SAT. What cutoff score should she use to select participants for her study?

What percent of the population obtains scores between 275 and 375?

What percent of the population obtains scores between 430 and 530?

Calculate protected t tests to compare all possible pairs of the four dormitories (e.g., Turck, Kirk, etc.), using the error term for Exercise 2 of the previous chapter. Which pairs differ significantly at the .05 level? Give two reasons why it would not be appropriate to calculate LSD for the data from the four dormitories.

Calculate LSD for the two experiments in Exercise 1 of the previous chapter. For which experiment is the calculation of LSD justified? Determine which pairs of means differ significantly in the experiment for which the calculation of LSD is justified.

**Data from exercise 1(previous chapter)**

Following are two separate (hypothetical) sets of data that are somewhat exaggerated to help clarify the procedures underlying analysis of variance. In each case, the experimenter is interested in conditions that affect the number of errors made by participants on a simple clerical task that must be performed quickly. Group 1 listens to upbeat dance music while performing the tasks; Group 2 listens to soothing, New Age music; and Group 3 listens to white noise (e.g., the steady hum of a machine in the background).

Using the means and standard deviations you have already calculated for the data from the four dormitories (i.e., Turck, Dirk, Dupre, and Doty Halls), perform an analysis of variance to decide whether you can reject the null hypothesis that the four samples come from populations with identical means. Calculate eta squared for this comparison, and comment on the proportion of variance accounted for by the different dormitories.

Recalculate the required sample sizes for part (a) of Exercise 7 if the two populations are matched with ρ = .6.

**Data from exercise 7**

a. Calculate g for the comparison of the means of Turck and Kirk Halls (if you did not already calculate that value for Exercise 1 of Chapter 7). Use g as your estimate of d, and compute the sample sizes you would need for power = .75, for a .05 two-tailed test, comparing the two dormitories.

Suppose that you plan to match students between Turck and Kirk Halls so that the population correlation corresponds to .4. Given the d you found in part (b) of Exercise 6 and the sizes of the samples from Turck and Kirk Halls, how much power would you have for a two-tailed matched-pairs t test at the .01 level?

If the sample r you calculated in Exercise 2 of Chapter 9 was equal to ρ for the population, how large a sample would you need to obtain power = .7 for a two-tailed test at the .05 level?

Calculate the power for the tests you conducted for Turck and Kirk Halls in Exercise 1 of Chapter 6. (Use the t values you calculated for that exercise as the values for δ in your power calculation.)

**Data from exercise 1 (chapter 6)**

Answer these problems by calculating a t value and comparing it to the critical value for a two-tailed test at the .05 level. (You can save time by using the means and standard deviations you calculated for the dormitories called Turck and Kirk Halls for previous exercises.)

The following data come from Exercise 5 in Chapter 7. Calculate r_{pb}for these data by assigning an X value of 0 to Group 1 and an X value of 1 to Group 2. What proportion of the variance in the scores is accounted for by group membership?

Use the data and your results from Exercise 1 in Chapter 9 to compute the linear regression equation for predicting the sophomore average (Y) from the number of cuts (C). What sophomore average would be predicted for a student who cut class eight times during the semester in question?

a. Convert the data in part (a) of Exercise 3 to ranks, separately for each variable, in order to compute the Spearman rank-order correlation coefficient. Test r_{S} for statistical significance by using Table I. Compare r_{S} to the Pearson r that you calculated for Exercise 3, part (a).

b. Redo part (a) for the data in part (c) of Exercise 3 (i.e., with X rearranged).

A psychology instructor develops a training method that is designed to improve the examination scores of poor students. The performance of a sample of 12 such students on the posttest (X_{1}) following training, and the pretest (X_{2}) prior to training, is shown.

Test the null hypothesis that the training method has no effect, using the Wilcoxon test.

What is the minimum score needed to rank in the top 5% of the population?

How do you explain the fact that the answer to Exercise 9 is smaller than the answer to Exercise 8, even though each problem deals with a 100-point range of scores?

What percent of the population obtains scores of 410 or less?

Find the mean and standard deviation (σ) for the following data set:

68.36, 15.31, 77.42, 84.00, 76.59, 68.43, 72.41, 83.05, 91.07, 80.62, 77.83

Transform each raw score in the preceding data set into a z score, a T score, and an SAT score. The remaining exercises are based on SAT scores, with the assumption that they follow the normal distribution with μ = 500 and σ = 100.

For data set 3 in Chapter 1 (reproduced here), transform each subject's (S) score on X to a z score. Then transform each subject's score on Y to a z score.

Which of each pair is better, or are they the same? (You should be able to answer by inspection.)

a. A T score of 47 and a z score of + 0.33

b. A T score of 64 and a z score of + 0.88

c. A T score of 42 and a z score of −1.09

d. A T score of 60 and a z score of + 1.00

e. A T score of 50 and a raw score of 26 (mean of raw scores = 26, σ = 9)

f. A z score of + 0.04 and a raw score of 1092 (mean of raw scores = 1113, σ = 137)

g. A z score of zero and a T score of 50

Consider the following data:

This student believes that he has performed better on the psychology test for two reasons: His score is higher, and he is only 8 points below average (as opposed to 10 points below average on the English test). Convert each of his test scores to a z score. Is he right?

For each of the following, compute the z score; then compute the T score.

a. A Turck Hall student with a score of 17.

b. A Turck Hall student with a score of 11.

c. A Kirk Hall student with a score of 17.

d. A Kirk Hall student with a score of 11.

e. A Dupre Hall student with a score of 6.

f. A Doty Hall student with a score of 6.

The mean of a set of scores is 8 and the standard deviation is 4. What will the new mean, standard deviation, and variance be if you:

a. Add 6.8 to every number?

b. Subtract 4 from every number?

c. Multiply every number by 3.2?

d. Divide every number by 4?

e. Add 6 to every number, and then divide each new number by 2?

Compute the biased and unbiased standard deviations for Distribution 6 in Table 3.2.

**Table 3.2**

Compute the range for each of the four dorms.

Using the raw data for Dupre Hall, demonstrate that the sum of the deviations from the mean equals zero.

Find the mode of Kirk Hall by inspecting the regular frequency distribution you created for that dorm when solving the first exercise of Chapter 2.

Use the same grouped distribution for this exercise that you used for Exercise 8.

a. Estimate the score that corresponds to the second decile.

b. Estimate the score that corresponds to the 50th percentile.

c. Estimate the score that corresponds to the 68th percentile.

Use the same distribution for this exercise that you used for Exercise 7.

a. Approximately what score corresponds to the 25th percentile?

b. Approximately what score corresponds to the 75th percentile?

Use the grouped frequency distribution that you created for Kirk Hall in Exercise 2 to solve this exercise.

a. Estimate the percentile rank corresponding to a score of 16.

b. Estimate the percentile rank corresponding to a score of 7.

Use the regular (ungrouped) frequency distribution that you created for Turck Hall in Exercise 1 to solve this exercise.

(a) Estimate the percentile rank corresponding to a score of 8.

(b) Estimate the percentile rank corresponding to a score of 12.

For each of the frequency distributions shown in the following table, state whether it is

a. (approximately) normal

b. Unimodal, skewed to the right

c. Unimodal, skewed to the left

d. Bimodal, approximately symmetric

e. Bimodal, skewed to the right

f. Bimodal, skewed to the left

g. (approximately) rectangular

h. J-curve

Using the results you found for Exercise 9 in Chapter 2, compute the interquartile range and SIQR for Turck Hall. Using the same methods, calculate these measures for Kirk Hall, as well.

Using the definitional formula, calculate the (biased) “variance” of Doty Hall in terms of deviations from the median, rather than the mean, of that distribution (just put the five scores in numerical order, and use the middle one as the median).

(a) How does this “variance” compare to the σ^{2} that you calculated in Exercise 7?

(b) What property of the variance from the mean is illustrated by this exercise?

Compute the median of Turck Hall from the regular frequency distribution you created for that dorm when solving the first exercise of Chapter 2.

Calculate the friendliness means of Turck, Kirk, Dupre, and Doty Halls (see first exercise of Chapter 1).

**Data from first exercise**

Compute the following (these values will not by themselves help Carrie to make a decision about the dormitories, but we will use these values as steps in future exercises to answer Carrie's concern):

(a) For Turck Hall:

(b) For Kirk Hall:

(c) For Dupre Hall:

(d) For Doty Hall:

Create a stem-and-leaf plot for the data from Turck Hall, using an interval size of 3.

Using the grouped frequency distributions you created for Turck and Kirk Halls in Exercise 2, plot the two corresponding frequency polygons on the same set of axes (i.e., in one graph).

Plot a histogram corresponding to the grouped frequency distribution that you created for Turck Hall in the previous exercise. What kind of distribution shape do you see?

Create grouped and cumulative grouped frequency distributions for Turck and Kirk Halls.

Create regular and cumulative frequency distributions for the data from the Turck, Kirk, and Dupre dormitories (see first exercise of Chapter 1).

**Data from first exercise**

Compute the following (these values will not by themselves help Carrie to make a decision about the dormitories, but we will use these values as steps in future exercises to answer Carrie's concern):

(a) For Turck Hall:

(b) For Kirk Hall:

(c) For Dupre Hall:

(d) For Doty Hall:

Compute the values needed to fill in the blanks.

For each of the following (separate) sets of data, compute the values needed in order to fill in the answer spaces. Then answer the additional questions that follow.

(Use the appropriate summation rule in each case so as to make the calculations easier.)

Five students are enrolled in an advanced course in psychology. Two quizzes are given early in the semester, each worth a total of 10 points. The results are as follows:

(a) Compute each of the following:

Î£X = ________ (Î£X)2 = ________ Î£(X âˆ’ Y) = ________

Î£Y = ________ (Î£Y)2 = ________ Î£X âˆ’ Î£Y = ________

Î£X2 = ________ Î£(X + Y) = ________ Î£XY = ________

Î£Y2 = ________ Î£X + Î£Y = ________ Î£X Î£Y = ________

(b) Using the results of part (a), show that each of the following rules listed in this chapter is true:

Rule 1: ________ = ________

Rule 2: ________ = ________

Rule 3: ________ â‰ ________

Rule 4: ________ â‰ ________ (X data)

________ â‰ ________ (Y data

(c) After some consideration, the instructor decides that Quiz 1 was excessively difficult and decides to add 4 points to each student's score. This can be represented in symbols by using k to stand for the constant amount in question, 4 points. Using Rule 6, compute

Compute

(Note that this result is different from the preceding one.)

Now add 4 points to each student's score on Quiz 1 and obtain the sum of these new scores.

(d) Had the instructor been particularly uncharitable, he might have decided that Quiz 2 was too easy and subtracted 3 points from each student's score on that quiz. Although this is a new problem, the letter k can again be used to represent the constant; here, k = 3.

Using Rule 7, compute

Compute

(Note that this result is different from the preceding one.)

Now subtract 3 points from each student's score on Quiz 2 and obtain the sum of these new scores.

(e) Suppose that the instructor decides to double all of the original scores on Quiz 1. Using Rule 8, compute

Now double each student's score on Quiz 1 and obtain the sum of these new scores.

Compute the following (these values will not by themselves help Carrie to make a decision about the dormitories, but we will use these values as steps in future exercises to answer Carrie's concern):

(a) For Turck Hall:

(a) For Turck Hall:

(b) For Kirk Hall:

(c) For Dupre Hall:

(d) For Doty Hall:

Express the following words in symbols.

(a) Add up all the scores on test X, then add up all the scores on test Y, and then add the two sums together.

(b) Add up all the scores on test G. To this, add the following: the sum obtained by squaring all the scores on test P and then adding them up.

(c) Square all the scores on test X. Add them up. From this, subtract 6 times the sum you get when you multiply each score on X by the corresponding score on Y and add them up. To this, add 4 times the quantity obtained by adding up all the scores on test X and squaring the result. To this, add twice the sum obtained by squaring each Y score and then adding them up.

(Compare the amount of space needed to express this equation in words with the amount of space needed to express it in symbols. Do you see why summation notation is necessary?)

Calculate the mean deviation for Doty Hall, and compare it to σ (as calculated in Exercise 7). What do you think would happen to the difference of these two measures if you were to add an extreme score to the data of Doty?

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