Question

An article summarized results from a national survey of 2,402 Americans age 8 to 18. The sample was selected to be representative of Americans in this age group. (Use a table or technology.) (a) of those surveyed, 1,125 reported owning a cell phone. Use this information to construct a 90% confidence interval estimate of the proportion of all Americans age 8 to 18 who own a cell phone. (Round your answers to three decimal places.) Interpret the interval. We are confident that the proportion of all Americans who own a cell phone is within this interval at least 90% of the time. We are confident that 90% of the proportion of all Americans age 8 to 18 who own a cell phone is within this interval. We are 90% confident that the proportion of all Americans who own a cell phone is within this interval. We are 90% confident that the proportion of all Americans age 8 to 18 who own a cell phone is within this interval. (b) of those surveyed, 1,621 reported owning an MP3 music player. Use this information to construct a 90% confidence interval for the proportion of all Americans age 8 to 18 who own an MP3 music player. (Round your answers to three decimal places.) Interpret the interval. We are 90% confident that the proportion of all Americans who own an MP3 player is within this interval. We are confident that 90% of the proportion of all Americans age 8 to 18 who own an MP3 player is within this interval. We are 90% confident that the proportion of all Americans age 8 to 18 who own an MP3 player is within this interval. We are confident that the proportion of all Americans age 8 to 18 who own an MP3 player is within this interval at least 90% of the time. (c) Explain why the confidence interval from part (b) is narrower than the confidence interval from part (a) even though the confidence levels and the sample sizes used to calculate the two intervals were the same. The sample proportion in part (b) is farther from O, thus increasing the value of the standard error of the sample proportion. The sample proportion in part (b) is farther from 0.5, thus reducing the value of the standard error of the sample proportion. The sample proportion in part (a) is farther from 0.5, thus reducing the value of the standard error of the sample proportion. The sample proportion in part (a) is farther from O, thus increasing the value of the standard error of the sample proportion.

Answer 1

a)

sample success x =		1125
sample size          n=		2402
sample proportion p̂ =x/n=		0.4680
std error se= √(p*(1-p)/n) =		0.0102
for 90 % CI value of z=		1.645
margin of error E=z*std error   =		0.0167
lower bound=p̂ -E                       =		0.4513
Upper bound=p̂ +E                     =		0.4847
from above 90% confidence interval for population proportion =(0.451,0.485)

(please try 0.452,0.485 if this comes wrong)

We are 90% confident that the proportion of all Americans age 8 to 18 who own a cell phone is within this interval.

b)

sample proportion p̂ =x/n=		0.6750
std error se= √(p*(1-p)/n) =		0.0096
for 90 % CI value of z=		1.645
margin of error E=z*std error   =		0.0157
lower bound=p̂ -E                       =		0.6593
Upper bound=p̂ +E                     =		0.6907
from above 90% confidence interval for population proportion =(0.659,0.691)

We are 90% confident that the proportion of all Americans age 8 to 18 who own an MP3 player is within this interval.

c)

The sample proportion in part (b) is farther from 0.5, thus reducing the value of the standard error of the sample proportion.