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Chapter 7: Problem 45

According to a 2017 Pew Research survey, \(60 \%\) of young Americans aged 18 to 29 say the primary way they watch television is through streaming services on the Internet. Suppose a random sample of 200 Americans from this age group is selected. a. What percentage of the sample would we expect to watch television primarily through streaming services? b. Verify that the conditions for the Central Limit Theorem are met. c. Would it be surprising to find that 125 people in the sample watched television primarily through streaming services? Why or why not? d. Would it be surprising to find that more than \(74 \%\) of the sample watched television primarily through streaming services? Why or why not?

Short Answer

Expert verified
a) 60%, b) The conditions are met as the sample size is sufficiently large and random, c) It would not be significantly surprising, as 125 is close to the expected 120, d) More than 74% or 148 individuals would indeed be surprising as it is significantly above the expected.

Step by step solution

01

Calculate Percentage

According to the problem, 60% of young Americans in the age group 18-29 watch TV primarily via streaming services. If a random sample of 200 is selected, we would expect this percentage to hold in the sample too. Thus, the percentage of the sample expected to watch television primarily through streaming services is 60%.
02

Verify the conditions for Central Limit Theorem

The Central Limit Theorem applies when the sample size is large (usually considered as over 30) and when the samples are taken randomly. Here, the sample size is 200 which is sufficiently large. We're also told the sample is random. Thus, the conditions for the Central Limit Theorem are met.
03

Analyze the number of people watching TV through streaming services

The expectation is that 60% of the sample, which means 120 people (0.60 * 200), watch TV primarily through streaming. If we find 125 people doing so, that is slightly higher but not far off the expected number, so it shouldn't be very surprising.
04

Assess the percentage of the sample watching TV through streaming services

A sample where more than 74% watch TV primarily through streaming would suggest at least 148 people (0.74 * 200) do so. This is significantly above our expected number (120), so it would indeed be surprising.

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