Question

13. [-/7 Points] DETAILS PECKSTAT2 9.R.090. MY NOTES ASK YOUR TEACHER An article summarized results from a national survey of 2,102 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, 1,226 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 90% confident that the proportion of all Americans who own a cell phone is within this interval. 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 age 8 to 18 who own a cell phone is within this 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. (b) of those surveyed, 1,722 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 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. 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 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 who own an MP3 player is within this interval. (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 0.5, thus reducing the value of the standard error of the sample proportion. 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 (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) 90% confidence interval : We have X 1226, N = 2102 P= 1226 2102 0.583 0.10 is ze The critical value for a = shown below: 1.645. The corresponding confidence interval is computed as Zc V Þ(1-P), Ô + ZcV p(1-P) N 0.583 – 1.645 * 0.0.583(1-0.0.583) 2102 0.583 +1.645 * 0.0.583(1-0.0.583) 2102 . (0.566, 0.601)

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

(b) 90% confidence interval : We have X = 1722, N = 2102 Ô= * 1722 0.819 The critical value for a = 0.10 is ze = 1.645. The corresponding confidence interval is computed as shown below: Þ(1-6), p + ZcV ZcV р p(1-ė) N N 0.819 1.645 * 0.0.819(1-0.0.819) 2102 0.819 + 1.645 * 0.0.819(1-0.0.819) 2102 (0.805, 0.833)

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.