The Daily Show. A 2010 Pew Research foundation poll indicates that among 1,099 c

Question

The Daily Show. A 2010 Pew Research foundation poll indicates that among 1,099 college graduates, 33% watch The Daily Show. Meanwhile, 22% of the 1,110 people with a high school degree but no college degree in the poll watch The Daily Show. A 95% confidence interval for (college grad - Phs or less), where p is the proportion of those who watch The Daily Show, is (0.07, 0.15). Based on this information, determine if the following statements are true or false, and explain your reasoning if you identify the statement as false.

(A)At the 5% significance level, the data provide convincing evidence of a difference between the proportions of college graduates and those with a high school degree or less who watch The Daily Show.

(B) We are 95% confident that 7% less to 15% more college graduates watch The Daily Show than those with a high school degree or less

(C)95% of random samples of 1,099 college graduates and 1,110 people with a high school degree or less will yield differences in sample proportions between 7% and 15%

(D)A 90% confidence interval for (pCollege grad - Phs or less) would be wider.

(E) A 95% confidence interval for (pns or less -Pcoiiege grad) is (-0.15,-0.07).

Explanation / Answer

Given,

Among 1,099 college graduates, 33% watch The Daily Show.

22% of the 1,110 people with a high school degree but no college degree in the poll watch The Daily Show.

95% confidence interval for (college grad - Phs or less) is  (0.07, 0.15)

(A)At the 5% significance level, the data provide convincing evidence of a difference between the proportions of college graduates and those with a high school degree or less who watch The Daily Show. - True

This is a true statement. As the 95% confidence interval of mean difference does not contain 0, at the 5% significance level, the data provide convincing evidence of a difference between the proportions of college graduates.

B) We are 95% confident that 7% less to 15% more college graduates watch The Daily Show than those with a high school degree or less - False

This is a false statement. We are 95% confident that 7% more to 15% more college graduates watch The Daily Show than those with a high school degree or less

(C) 95% of random samples of 1,099 college graduates and 1,110 people with a high school degree or less will yield differences in sample proportions between 7% and 15% - True

This is a true statement. As the 95% confidence interval for (college grad - Phs or less) is  (0.07, 0.15), 95% of random samples of college graduates and people with a high school degree or less will yield differences in sample proportions between 7% and 15%

(D)A 90% confidence interval for (pCollege grad - Phs or less) would be wider. - False

This is a false statement. The Z value of 90% is 1.64 which is less than the Z value of 95% (1.96). So, margin of error at 90% confidence interval is less than that at 95%. So, width of confidence interval which is dependent on margin of error will bae narrower than width of 95% confidence interval.

(E) A 95% confidence interval for (pns or less -Pcoiiege grad) is (-0.15,-0.07). - True

This is a true statement. Let the Differenece in means of proportions of (college grad - Phs or less) be X.

Let M be margin of error at 95% confidence interval. Then,

X - M = 0.07 and X+ M = 0.15

=> X = 0.11 and M = 0.04

Now, 95% confidence interval for (pns or less -Pcoiiege grad) is

(-X - M, -X + M)

= (-0.11 - 0.04, -0.11 + 0.04)

= (-0.15, -0.07)

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