Problem set 1: Use the following information to answer questions 1-5 (20 points):

A criminologist developed a test to measure recidivism, where low scores indicated a lower probability of repeating the undesirable behavior. The test is normed so that it has a mean of 125 and a standard deviation of 35.

1.What is the percentile rank of a score of 180 (4 pts)?

2.What is the Z score for a test score of 55 (4 pts)?

3.What percentage of scores falls between 90 and 177.5 (4 pts)?

4.What proportion of respondents should score above 185 (4 pts)?

5.Suppose an individual is in the 75.8th percentile in this test, what is his or her corresponding recidivism score (4 pts)?

Problem set 2: Use the following information to answer questions 6-14 (40 points):

The mean age at marriage for respondents in the 2020 General Social Survey (GSS) is 25.33, with a standard deviation of 6.76.

6.From 2020 GSS, calculate the z-score associated with an observed age at first marriage of 27 and provide a substantive interpretation of this quantity (5 pts)?

7.From 2020 GSS, calculate the z-score associated with an observed age at first marriage of 18 and provide a substantive interpretation of this quantity (5 pts)?

8.From 2020 GSS, assume that the z-score associated with a particular age at first marriage is 0.52. If the proportion of the area between this particular age at first marriage and the mean is 0.1985, what proportion of respondents experienced their first marriage earlier than this age (5 pts)?

9.From 2020 GSS, calculate the observed age at marriage associated with a z-score of -0.99 (5 pts).

10.From 2020 GSS, suppose that a person experienced their first marriage at age 23. If the area beyond the z-score associated with age 23 is 0.3669, what proportion of respondents experienced their first marriage before age 23 (5 pts)?

11.From 2020 GSS, suppose that a person experienced their first marriage at age 23. If the area beyond the z-score associated with age 23 is 0.3669, what proportion of respondents experienced their first marriage after age 23 (5 pts)?

12.From 2020 GSS, suppose that the proportion of area between the mean and two z-score of +0.42 and -0.42 are both 0.1628. Calculate the raw scores associated with these two z-scores. What proportion of respondents were first married between these two ages (3 pts)?

13.From 2020 GSS, the z-score associated with the top 5 percent of the distribution is approximately 1.65. What is the observed age at first marriage associated with this z-score (5 pts)?

14.From 2020 GSS, for a first age at marriage of 36.48, the proportion of area beyond the z-score associated with this age is 0.0495. What is the percentile rank for this score (2 pts)?

Problem set 3: use the following information to answer questions 15-19 (15 pts total):

Based on the SPSS Demonstration on page 169, we find the mean number of years of education is 13.71 with a standard deviation of 3.039 round to 3.04. A total of 1,498 GSS 2018 respondents were included in the survey. Assuming that years of education is normally distributed, answer the following questions.

15.If you have 13.71 years of education, that is, the mean number of years of education, what is your Z score (3pts)?

16.If your friend is in the 67th percentile, how many years of education does she have (3pts)?

17.How many people have between your years of education (13.71) and your friend's years of education (3pts)?

18.Concerning the data in 2018 GSS, the standard normal table reports the following information in the table below. Calculate the proportion of respondents who had 14 years of education (3pts).

Education Proportion of Area Between Mean and Z Proportion of Area Beyond Z

14 .0398 .4602

15 .1628 .3372

19. Concerning the data in 2018 GSS, the standard normal table reports the following information in the table below.Calculate the proportion of respondents who had between 14 and 15 years of education (3pts).

Education Proportion of Area Between Mean and Z Proportion of Area Beyond Z

14 .0398 .4602

15 .1628 .3372

Problem set 4: use the following table to answer questions 20-23 (25 pts total):

In this table, We report the average years of education for a subsample of GSS 2018 respondents by their social classlower, working, middle, and upper. Standard deviations are also reported for each class.

Mean Standard Deviation N

Lower class 12.03 2.93 142

Working class 13.05 2.85 541

Middle class 14.56 2.62 475

Upper class 15.48 2.33 34

20.Assuming that years of education is normally distributed in the population, what proportion of working-class respondents have 10 to 15 years of education (5 pts)? What proportion of middle-class respondents have 10 to 15 years of education (5 pts)?

21.What is the probability that a working-class respondent, drawn at random from the population, will have more than 16 years of education (5 pts)? What is the equivalent probability for a lower-class respondent drawn at random (5 pts)?

22.What is the probability that an upper-class respondent will have less than 10 years of education (3 pts)?

23.If years of education is positively skewed in the population, how would that change your other answers (2 pts)?