Fall 2015 Statistics for Social Sciences Anderson 1 Problem Set #2 This problem set covers Chapters 5 – 8 of your text and is due at the start of class on Tuesday, November 17. Please write neatly and show all your work where applicable so that I can give you partial credit! 1. You’ve developed a new aptitude test, which you’ve named the Scholastic Math And Reading Test (SMART). You gave your test to a population of consisting of all the high school seniors in a small rural community. This population had a mean of 77.25 and a standard deviation 12.81. a. One student took the SMART and received an 87. Calculate her z-score. (Report to two decimal places.) b. You’d like to be able to compare scores on the SMART with another standardized aptitude tests–the ACT. In order to do this, we need to transform the SMART distribution so that it has the same mean and standard deviation as the ACT. The ACT has a mean of 20 and standard deviation of 5. If a students scored an 87 on the SMART, what would her score be in the new, transformed distribution? (Report to one decimal place.) c. Suppose we transformed the ACT so that it had the same mean and standard deviation as the SMART. A 30 on the ACT is comparable to what score on the SMART? (Report to two decimal places.) d. Which score is better: an 18 on the ACT or a 70 on the SMART? Explain how you got your answer. 2. The following table summarizes the number of individuals that that were offered special assistance in three areas by a community agency: Drugs/Alcohol 10 Family crisis counseling 20 Other 20 a. If you were to select someone at random from the records for last year, what is the probability that the person would be in the “drugs/alcohol” category? (Please respond in decimal form.) b. If you were to select someone at random from the records for last year, what is the probability that the person would NOT be in the “Other” category? (Please respond in decimal form,) Fall 2015 Statistics for Social Sciences Anderson 2 3. Over the past 10 years, the local school district has measured physical fitness for all high school freshmen. During that time, the mean score on a treadmill endurance task has been 19.8 minutes with a standard deviation of 7.2 minutes. Assume that the distribution is approximately normal. a. What is the probability of randomly selecting a student with a treadmill time greater than 25 minutes? (Please report as a percentage with 2 decimal places) b. Please explain, step-by-step, how you went about answering Question 3a. c. If the school required a minimum time of 10 minutes for students to pass the physical education course, what percentage of the freshmen would fail? (Please give two decimal places) 4. Assume you have a normally distributed population with a µ = 6.8 and a s = 12. a. Taking the mean of each and every sample of n = 4 from the population would give you a distribution known as the ________________. b. What is the mean of the distribution created in Question 4a? c. The mean you reported in Question 4b is also known as the ________________. d. What is the standard deviation of the distribution we created in Question 4a? e. The standard deviation you calculated in Question 4d is also known as the ____. f. Given the information from Question 4, what is the probability of randomly selecting a sample of n = 4 scores with a mean less than 8? (Please report as a percentage to two decimal places.) g. Suppose that rather than being normally distributed, the population described in Question 10 was severely negatively skewed. Explain why you would not be able to find the probability of obtaining a sample mean less than M = 8 using the methods we’ve covered in this class. Fall 2015 Statistics for Social Sciences Anderson 3 5. Suppose UMM is evaluating a new freshman composition course. A random sample of 25 freshmen is obtained and the students are placed in the course during their first semester. One year later, a writing sample is obtained for each student, and the writing samples are graded using a standardized evaluation technique. The average score for the sample is 76. For the general population of college students, writing scores on the standardized evaluation are normally distributed and have a mean of 70 and standard deviation of 20. a. Assuming a two-tailed test, state the null and alternative hypotheses. Please give me a one sentence verbal description of each (i.e. tell me the hypothesis in plain English) AND the mathematical notation. b. Define “critical region.” Assuming an alpha of .05, where is the critical region for this z test? c. What is the z-score for the sample described in Question 5? d. What decision should be made about the null hypothesis and why? What does this suggest about the effectiveness of the new composition course? e. Calculate Cohen’s d for the treatment effect. f. Explain what researchers mean by the “power” of a hypothesis test. Assuming that all other factors are held constant how would each of the the following changes below affect the power of the test we just conducted? i. Changing the alpha level from .05 to .01 ii. Changing from a two-tailed test to a one-tailed test 6. Two researchers are studying treatments for bipolar disorder: • Maria’s hypothesis test led her to conclude that her treatment did not have an effect when, in reality, it did. • Robert’s hypothesis test led him to conclude that his treatment had an effect when, in reality, it did not. a. What type of error did Maria make? b. What type of error did Robert make? c. Who made the more serious error? Explain. d. How can Robert ensure that he only has a 1% chance of making the same type of error in his next study?

Fall 2015 Statistics for Social Sciences Anderson 1 Problem Set #2 This problem set covers Chapters 5 – 8 of your text and is due at the start of class on Tuesday, November 17. Please write neatly and show all your work where applicable so that I can give you partial credit! 1. You’ve developed a new aptitude test, which you’ve named the Scholastic Math And Reading Test (SMART). You gave your test to a population of consisting of all the high school seniors in a small rural community. This population had a mean of 77.25 and a standard deviation 12.81. a. One student took the SMART and received an 87. Calculate her z-score. (Report to two decimal places.) b. You’d like to be able to compare scores on the SMART with another standardized aptitude tests–the ACT. In order to do this, we need to transform the SMART distribution so that it has the same mean and standard deviation as the ACT. The ACT has a mean of 20 and standard deviation of 5. If a students scored an 87 on the SMART, what would her score be in the new, transformed distribution? (Report to one decimal place.) c. Suppose we transformed the ACT so that it had the same mean and standard deviation as the SMART. A 30 on the ACT is comparable to what score on the SMART? (Report to two decimal places.) d. Which score is better: an 18 on the ACT or a 70 on the SMART? Explain how you got your answer. 2. The following table summarizes the number of individuals that that were offered special assistance in three areas by a community agency: Drugs/Alcohol 10 Family crisis counseling 20 Other 20 a. If you were to select someone at random from the records for last year, what is the probability that the person would be in the “drugs/alcohol” category? (Please respond in decimal form.) b. If you were to select someone at random from the records for last year, what is the probability that the person would NOT be in the “Other” category? (Please respond in decimal form,) Fall 2015 Statistics for Social Sciences Anderson 2 3. Over the past 10 years, the local school district has measured physical fitness for all high school freshmen. During that time, the mean score on a treadmill endurance task has been 19.8 minutes with a standard deviation of 7.2 minutes. Assume that the distribution is approximately normal. a. What is the probability of randomly selecting a student with a treadmill time greater than 25 minutes? (Please report as a percentage with 2 decimal places) b. Please explain, step-by-step, how you went about answering Question 3a. c. If the school required a minimum time of 10 minutes for students to pass the physical education course, what percentage of the freshmen would fail? (Please give two decimal places) 4. Assume you have a normally distributed population with a µ = 6.8 and a s = 12. a. Taking the mean of each and every sample of n = 4 from the population would give you a distribution known as the ________________. b. What is the mean of the distribution created in Question 4a? c. The mean you reported in Question 4b is also known as the ________________. d. What is the standard deviation of the distribution we created in Question 4a? e. The standard deviation you calculated in Question 4d is also known as the ____. f. Given the information from Question 4, what is the probability of randomly selecting a sample of n = 4 scores with a mean less than 8? (Please report as a percentage to two decimal places.) g. Suppose that rather than being normally distributed, the population described in Question 10 was severely negatively skewed. Explain why you would not be able to find the probability of obtaining a sample mean less than M = 8 using the methods we’ve covered in this class. Fall 2015 Statistics for Social Sciences Anderson 3 5. Suppose UMM is evaluating a new freshman composition course. A random sample of 25 freshmen is obtained and the students are placed in the course during their first semester. One year later, a writing sample is obtained for each student, and the writing samples are graded using a standardized evaluation technique. The average score for the sample is 76. For the general population of college students, writing scores on the standardized evaluation are normally distributed and have a mean of 70 and standard deviation of 20. a. Assuming a two-tailed test, state the null and alternative hypotheses. Please give me a one sentence verbal description of each (i.e. tell me the hypothesis in plain English) AND the mathematical notation. b. Define “critical region.” Assuming an alpha of .05, where is the critical region for this z test? c. What is the z-score for the sample described in Question 5? d. What decision should be made about the null hypothesis and why? What does this suggest about the effectiveness of the new composition course? e. Calculate Cohen’s d for the treatment effect. f. Explain what researchers mean by the “power” of a hypothesis test. Assuming that all other factors are held constant how would each of the the following changes below affect the power of the test we just conducted? i. Changing the alpha level from .05 to .01 ii. Changing from a two-tailed test to a one-tailed test 6. Two researchers are studying treatments for bipolar disorder: • Maria’s hypothesis test led her to conclude that her treatment did not have an effect when, in reality, it did. • Robert’s hypothesis test led him to conclude that his treatment had an effect when, in reality, it did not. a. What type of error did Maria make? b. What type of error did Robert make? c. Who made the more serious error? Explain. d. How can Robert ensure that he only has a 1% chance of making the same type of error in his next study?