stats: chapter 7 practice test

young players in the NFL, the distribution of age is skewed to the right with a mean of 26.2 years and a standard deviation of 3.24 years. in the NBA, the distribution of age is skewed to the right with a mean of 25.8 years and a standard deviation of 4.24 years. independent random samples of 20 players in each league are selected. let x-bar nfl represent the sample mean age of nfl players and let x-bar nba represent the sample mean age of nba players (a) find the mean of the sampling distribution of x-bar nfl - x-bar nba

Mx nfl - x-bar nba = M nfl - M nba = 26.2 - 35.8 = 0.4 years

a study of college freshmen's study habits found that the time (in hours) that freshmen study each week follows a distribution with a mean of 7.2 hours and a standard deviation of 5.3 hours (c) find the probability that the average number of hours spent studying by an srs of 55 students is greater than 9 hours

x-bar = average # of hours spent studying by a srs of 55 students where distribution of x-bar is approx N(7.2, 0.7146) P(x-bar > 9) = P(z > 2.5188) = 0.0058 normalcdf (LB: 9, UB: infinity, m: 7.2, sd: 0.7146) z = 9-7.2 / 0.7146 = 2.5188 there is a 0.0058 probability that the average # of hours spent studying by a srs of 55 students is greater than 9 hours

an opinion poll asks a sample of 500 adults (an srs) whether they favor giving parents of school-age children vouchers that can be exchanged for education at any public or private school of their choice. each school would be paid by the government on the basis of how many vouchers it collected. suppose that in fact 45% of the population favor this idea. (a) determine the mean and standard deviation of the sampling distribution of p-hat, the proportion of adults in samples of 500 who favor giving parents of school-age children these vouchers?

mp = p = 4.5 since there are at least 500(10) = 5000 adults, the 10% condition is satisfied therefore, op = sq rt p(1-p)/n = sq rt (0.45)(0.55)/500 = 0.0222

a researcher reports that 80% of high school graduates, but only 40% of high school dropouts, would pass a basic literacy test. assume that the researcher's claim is true. suppose we give a basic literacy test to a random sample of 60 high school graduates and an independent random sample of 75 high school dropouts. let p-hat g and p-hat d be the sample proportions of graduates and dropouts, respectively, who pass the test (b) find the mean of the sampling distribution of pg - pd

mpg - pd = pg - pd = 0.8 - 0.4 = 0.4

a study of college freshmen's study habits found that the time (in hours) that freshmen study each week follows a distribution with a mean of 7.2 hours and a standard deviation of 5.3 hours (b) what are the mean and standard deviation of the sampling distribution of the mean for samples of 55 randomly selected freshmen?

mx = m = 7.2 hours since there are at least 55(10)=550 college freshmen, the 10% condition is satisfied so ox = o/sq rt n = 5.3/sq rt 55 = 0.7146 hours

a study of college freshmen's study habits found that the time (in hours) that freshmen study each week follows a distribution with a mean of 7.2 hours and a standard deviation of 5.3 hours (a) what is the shape of the sampling distribution of the mean x-bar for samples of 55 randomly selected freshmen?

the sampling distribution of x-bar is approx normal because n = 55 .= 30 ; guaranteed by CLT

a researcher reports that 80% of high school graduates, but only 40% of high school dropouts, would pass a basic literacy test. assume that the researcher's claim is true. suppose we give a basic literacy test to a random sample of 60 high school graduates and an independent random sample of 75 high school dropouts. let p-hat g and p-hat d be the sample proportions of graduates and dropouts, respectively, who pass the test (a) what is the shape of p-hat g - p-hat d? why?

the shape of the sampling distribution of p-hat g - p-hat d is approx normal because large condition is satisfied since ngpg = 60(0.8) = 48, ng(1-pg) = 60(0.2) = 12 >= 10 and ndpd = 75(0.4) = 30, nd(1-pd) = 75(0.6) = 45 >= 10

young players in the NFL, the distribution of age is skewed to the right with a mean of 26.2 years and a standard deviation of 3.24 years. in the NBA, the distribution of age is skewed to the right with a mean of 25.8 years and a standard deviation of 4.24 years. independent random samples of 20 players in each league are selected. let x-bar nfl represent the sample mean age of nfl players and let x-bar nba represent the sample mean age of nba players (c) is the shape of the sampling distribution approximately normal? justify your answer

no, since it stated that both population distributions were skewed to the right, the sample sizes must be >= 30 in order for CLT to guarantee normality. in this case, both sample sizes are 20 which is < 30 so we cannot assume the sampling distribution is approx normal

an opinion poll asks a sample of 500 adults (an srs) whether they favor giving parents of school-age children vouchers that can be exchanged for education at any public or private school of their choice. each school would be paid by the government on the basis of how many vouchers it collected. suppose that in fact 45% of the population favor this idea. (c) what is the probability that more than half of the sample are in favor?

p-hat = proportion of adults in sample of srs that favor giving parents of school aged children vouches that can be exchanged at school of their choice where the distribution of p-hat is approx N(0.45, 0.02220 p(p-hat > 0.5) = p(z > 2.2522) = 0.0121 normalcdf(LB: 2.2322, UB: infinity, m: 0, sd: 1) z = 0.5-0.45/0.0222 = 2.2322 there is a 0.0121 probability that more than half of the sample are in favor

a researcher reports that 80% of high school graduates, but only 40% of high school dropouts, would pass a basic literacy test. assume that the researcher's claim is true. suppose we give a basic literacy test to a random sample of 60 high school graduates and an independent random sample of 75 high school dropouts. let p-hat g and p-hat d be the sample proportions of graduates and dropouts, respectively, who pass the test (d) find the probability that the proportion of graduates who pass the test is at most 0.20 higher than the proportion of dropouts who pass, assuming that the researcher's report is correct

pg - pd = difference in sample proportions of those who would pass a basic literacy test where the distribution of pg - pd is approx N(0.4, 0.0765) p(pg - pd <= 0.2) = p(z <= -2.6143) = 0.0044 normalcdf (LB: negative infinity, UB: -2.6143, m: 0, sd: 1) z = 0.2-0.4/0.0765 = -2.6143 there is a 0.0044 probability that the proportion of graduates who pass the test is at most 0.20 higher than the proportion of dropouts who pass the test, assuming that the researchers report is correct

an opinion poll asks a sample of 500 adults (an srs) whether they favor giving parents of school-age children vouchers that can be exchanged for education at any public or private school of their choice. each school would be paid by the government on the basis of how many vouchers it collected. suppose that in fact 45% of the population favor this idea. (b) is it reasonable to assume that the sampling distribution of p-hat is approximately normal?

since np = 500(0.45) = 225 and n(1-p) = 500(0.55) = 275 is >= 10, large counts condition is satisfied therefore sampling distributions of p-hat is approx normal

young players in the NFL, the distribution of age is skewed to the right with a mean of 26.2 years and a standard deviation of 3.24 years. in the NBA, the distribution of age is skewed to the right with a mean of 25.8 years and a standard deviation of 4.24 years. independent random samples of 20 players in each league are selected. let x-bar nfl represent the sample mean age of nfl players and let x-bar nba represent the sample mean age of nba players (b) calculate the standard deviation of the sampling distribution of x-bar nfl - x-bar nba

since there are at least 20(10) = 200 players in both the nfl and nba, the 10% condition is satisfied and it was stated that the samples were independent random samples, therefore O x-bar nfl - x-bar nba = sq rt O^2 nfl/n nfl + O^2 nba/n nba = sq rt 3.24^2/20 + 4.24^2/20 = 1.1932 years

a researcher reports that 80% of high school graduates, but only 40% of high school dropouts, would pass a basic literacy test. assume that the researcher's claim is true. suppose we give a basic literacy test to a random sample of 60 high school graduates and an independent random sample of 75 high school dropouts. let p-hat g and p-hat d be the sample proportions of graduates and dropouts, respectively, who pass the test (c) calculate the standard deviation of the sampling distribution of pg - pd?

since there are at least 60(10) = 600 high school graduates and 75(10) = 750 high school drououts the 10% condition is satisfied and it stated the samples were independent random samples, therefore o pg - pd = sq rt pg (1-pg)/ng + pd (1-pd)/nd = sq rt (0.8)(0.2)/60 + (0.4)(0.6)/75 = 0.0765