1. A social psychologist has developed a test measure gregariousness. The test is normed so that it has a mean of 70 and a standard deviation of 20, and the gregariousness scores are normally distributed in the population of college students used to develop the test.
a. What is the percentile rank of a score of 40?
b. What percentage of the scores fall between 35 and 90?
c. What is the standard score for a test score of 65?
d. What proportion of students should score above 115?
e. What is the cut off score below which 87% of all scores fall?
2. You are asked to do a study of shelters for abused and battered women to determine the necessary capacity in your city to provide housing for most of these women. After recording data for a whole year, you find that the mean number of women in shelters each night is 250, with a standard deviation of 75. Fortunately, the distribution of the number of women in the shelters each night is normal, so you can answer the following questions posed by city council.
a). If the city's shelters have a capacity of 350, will that be enough places for abused women on 95 percent of all nights? If not, what number of shelter openings will be needed?
b). The current capacity is only 220 openings because some shelters have closed. What is the percentage of nights that the number of abused women seeking shelter will exceed current capacity?
This solution shows step-by-step calculations to determine the percentile rank, standard score, proportion, cut off scores, and accepted capacity. Normal distribution diagrams are shown in each calculation for better understanding.