Question
You are going to create a 95% confidence interval for a population proportion and want the margin of error to be no more than 0.05. Historical data indicate that the population proportion has remained constant at about 0.7. What is the minimum size random sample you need to construct this interval?
a. 385
b. 322
c. 274
d. 275
e. 323
▶️Answer/Explanation
Ans:The correct answer is (e).
$
\begin{aligned}
& p=0.7, M=0.05, z^*=1.96(\text { for } C=0.95) \Rightarrow \\
& n \geq\left(\frac{z^*}{M}\right)^2(\hat{p})(1-\hat{p})=\left(\frac{1.96}{0.05}\right)^2(0.7)(0.3)=322.7 \text {. You need a sample of at least } \\
& n=323 .
\end{aligned}
$
Question
Which of the following would result in a smaller margin of error in a confidence interval for a mean?
a. Increasing the sample size
b. Increasing the confidence level
c. More variability in the sample responses
d. A smaller sample mean
e. All of these would result in a smaller margin of error.
▶️Answer/Explanation
Ans:The correct answer is (a).
Options (b) and (c) would increase the margin of error, and option (d) would not change it.
Question
St. Norbert College in Green Bay conducts an annual survey of Wisconsin residents. In 2011 a random sample of 400 Wisconsin residents was selected and \(25 \%\) of respondents said the country is headed in the right direction. In a 2016 random sample of 664 residents, \(28 \%\) said they thought the country was headed in the right direction. Do these samples provide convincing evidence of a change in the proportion of adults who thought the country was headed in the right direction?
a. No, because the \(95 \%\) confidence interval for the difference contains \(3 \%\).
b. No, because the \(95 \%\) confidence interval for the difference contains \(0 \%\).
c. No, because the \(95 \%\) confidence interval for the difference does not contain \(0 \%\).
d. Yes, because the \(95 \%\) confidence interval for the difference does not contain 3\%.
e. Yes, because the \(95 \%\) confidence interval for the difference contains \(3 \%\).
▶️Answer/Explanation
Ans:The correct answer is (b).
It is not necessary to calculate the interval to answer this question because (b) is the only choice that is logically consistent. The question is whether \(0 \%\) is in the interval, which leaves only options (b) and (c). And if 0 is in the interval, then there is not convincing evidence of a change. And the interval actually does contain 0 .
Question
You are going to construct a \(90 \% \mathrm{t}\) confidence interval for a population mean based on a sample size of 16 . What is the critical value of \(t\left(t^*\right)\) you will use in constructing this interval?
a. 1.341
b. 1.753
c. 1.746
d. 2.131
e. 1.337
▶️Answer/Explanation
Ans: The correct answer is (b).
\(n=16 \Rightarrow d f=16-1=15\). Using a table of \(t\) distribution critical values, look in the row for 15 degrees of freedom and the column with 0.05 at the top (or \(90 \%\) at the bottom). On a TI-83/84 with the invT function, the solution is given by invT \((0.95,15)\).