The General Social Survey (GSS) is a sociological survey used to collect data on demographic characteristics and attitudes of residents of the United States. The 2010 General Social Survey asked the question, "For how many days during the past 30 days was your mental health not good?" Based on responses from 1,151 US adults, the survey reported a \(95 \%\) confidence interval of \((3.40,4.24)\) days. Assume that the sampled US adults are representative of all US adults.

(a) Identify each of the following statements as true or false. Justify your answers.

i. The confidence interval of \((3.40,4.24)\) contains the mean days out of the past 30 days that U.S. adults experienced poor mental health.

ii. There is a \(95 \%\) chance that the mean days out of the past 30 days that U.S. adults experienced poor mental health is within the confidence interval \((3.40,4.24)\).

iii. If we repeated this survey 1,000 times and constructed a \(95 \%\) confidence interval each time, then approximately 950 of those intervals would contain the true mean days out of the past 30 days that U.S. adults experienced poor mental health.

iv. The survey provides statistically significant evidence at the \(\alpha=0.05\) significance level that the mean days out of the past 30 days that U.S. adults experienced poor mental health is not 4.5 days.

v. We can be \(95 \%\) confident that the mean days out of the past 30 days that U.S. adults experienced poor mental health is 3.82 days.

vi. We can be \(95 \%\) confident that the interval \((3.40,4.24)\) days contains the mean days out of the past 30 days that the sampled adults experienced poor mental health.

(b) Would you expect the \(90 \%\) confidence interval to be larger or smaller than the \(95 \%\) confidence interval? Explain your reasoning.

(c) Calculate the \(90 \%\) confidence interval.