statistics exam #2 questions
Based on the SPSS demonstration, we find the mean number of years of education is 13.77 with a standard deviation of 3.07. A total of 1,500 GSS respondents were included in the survey. a) if you have 13.77 years of education, that is, the mean number of education, what is your z-score? b) if your friend is in the 60th percentile, how many years of education does she have? c) how many people have between your years of education and your friend's years of education?
a) 13.77-13.77/3.07 = 0 b) 14.54 years 60/100 = 0.6 1.0-0.6 = 0.4 0.4 = 0.25 13.77+0.25(3.07) = 14.5375 = 14.54 c) 150 people have between 13.77 and 14.54 years of education. 1,500 x 0.10 = 150
A criminologist developed a test to measure recidivism, where low scores indicated a lower probability of repeating undesirable behavior. The test is normed so it has a mean of 140 and a standard deviation of 40. a) what is the percentile rank of a score of 172? b) what is the z-score for a test score of 200? c) what percentage of scores fall between 100 and 160? d) suppose an individual is in the 67th percentile in this test, what is his or her corresponding recidivism score?
a) 78.81st percentile 172-140/40=0.80 0.80 = 0.2119 1-0.2119 = 0.7881 0.7881 x 100 = 78.81 b) 200-140/40 = 1.5 c) 53.28% of scores fall between 100 and 160 100-140/40 = -1 the area between mean and z is 0.3413 160-140/40 = 0.5 the area between mean and z is 0.1915 0.3413+0.1915=0.5328 0.5328 x 100 = 53.28 d) their recidivism score is 157.6. 67/100 = .67 1-.67 = 0.33 0.33=0.44 y = 140+0.44(40) = 157.6
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. a) if the city's shelters have a capacity of 350, will that be enough places for abused women on 95% of nights? if not, what number of shelter openings will be needed? b) the current capacity if 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?
a) the 95th percentile corresponds to a z-score pf about 1.65. 95/100 = 0.95 1.00-0.95 = 0.05 closest number to 0.05 in column c = 0.0495 z score = 1.65 y = 250+1.65(75) = 373.75 women there is not enough space for all abused women on 95% of nights. the city needs at least 374 beds. b) 220-250/75 = -0.40 the area below this value is 0.3446, so the area exceeding this z is 1-.3446=0.6554, or 65.54% of all nights the number of women seeking shelter will exceed the capacity of 220.
we report the average years of education for a subsample of GSS 2014 respondents by their social class. table1 a) assuming that years of education is normally distributed in the population, what proportion of the working-class respondents have 12 to 16 years of education? what proportion of upper-class respondents have 12 to 16 years of education? b) what is the probability that a working-class respondent, drawn at random from the population, will have more than 16 years of education? what is the equivalent probability for a middle-class respondent drawn at random? c) what is the probability that a lower-class respondent will have less than 10 years of education? d) if years of education is actually positively skewed in the population, how would that change your other answers?
a) working class: 12-13.01/2.91 = -0.35, area between the mean and z is .1368. 16-13.01/2.91 = 1.03, area between mean and z is .3485 .1368+.3485=0.4853 the proportion of working-class respondents with 12 to 16 years of education is 0.4853 upper class: 12-15.44/2.83 = -1.22, area between the mean and z is 0.3888 16-15.44/2.83 = 0.2, area between mean and z is 0.0793 the proportion of upper-class respondents with 12 to 16 years of education is 0.4681 the working class has a higher proportion of respondents with 12-16 years of education than the upper class b) working class: 16-13.01/2.91 = 1.03, area beyond z is 0.1515 middle class: 16-14.99/2.93 = 0.35, area beyond z is 0.3632 the probability of a working-class respondent having more than 16 years of education is 15.15%. the probability of a middle-class respondent having more than 16 years of education is 36.32% c) 10-12.11/2.83 = -0.75, area beyond z is 0.2266. the probability of a lower-class respondent having 10 years of education is 22.66%. d) if years of education is positively skewed, then the proportion of cases with high levels of education will be less than that for a normal distribution. for example, the probabilities associated with high levels of education will be smaller.
We discovered that 1,001 GSS 2014 respondents watched television for an average of 2.94 hours per day with a standard deviation of 2.60 hours. a) what is the z-score for a person who watches more than 8 hours per day? b) what proportion of people watch television less than 5 hours a day? How many does this correspond to in the sample? c) what number of television hours per day corresponds to a z score of +1? d) what is the percentage of people who watch between 1 and 6 hours of television per day?
a) z = 8-2.94/2.60 = 1.95 b) z = 5-2.94/2.60 = 0.79 the area between the z and the mean is 0.2852. add 0.5+0.2852 to find the proportion. the proportion of people who watch television for less than 5 hours a day is 0.7852. This corresponds to 786 people (0.7852 x 1,001 = 785.99). c) y = 2.94+1(2.60) = 5.54 5.54 tv hours corresponds to a z-score of +1. d) z = 1-2.94/2.60 = -.75 the z-score for a person who watches 1 hour of tv per day is -.75. the area between the mean and the z is 0.2734. z = 6-2.94/2.60 = 1.18 the z-score for a person who watches 6 hours of tv per day is 1.18. the area between the mean and the z is 0.3810. 0.2734+0.3810=0.6544 0.6544 x 100 = 65.44 The percentage of people who watch between 1 and 6 hours of tv per day is 65.44%.
