Chapter 9.2, Problem 54E

(a)

To determine

To construct: and interpret a 95% confidence interval for the true proportion p all first year students at the university who would identify being very well-off as an important personal goal.

Answer to Problem 54E

  $0.5943<p<0.7257$

There is 95% confident that the true proportion of all first- year students at the university who would identify being very well-off as an important personal goal is between 0.5943 and 0.7357.

Explanation of Solution

Given:

  $\begin{array}{c}n=200\\ p=73\%=0.73\\ x=132\end{array}$

Formula used:

  $\begin{array}{c}\widehat{p}=\frac{x}{n}\\ E=z_{\alpha /2}\cdot \sqrt{\frac{\widehat{p}\left(1-\widehat{p}\right)}{n}}\end{array}$

Calculation:

The sample proportion is

  $\widehat{p}=\frac{x}{n}=\frac{132}{200}=0.66$

For confidence level $1-\alpha =0.95$ , determine $z_{\alpha /2}=z_{0.025}$ using table II (look up 0.025 in the table, the z-score is then they found z-score with opposite sign):

  $z_{\alpha /2}=1.96$

The margin of error is

  $\begin{array}{c}E=z_{\alpha /2}\cdot \sqrt{\frac{\widehat{p}\left(1-\widehat{p}\right)}{n}}\\ =1.96\times \sqrt{\frac{0.66\left(1-0.66\right)}{200}}\\ =0.0657\end{array}$

The confidence interval then becomes:

  $\begin{array}{c}\widehat{p}-E<p<\widehat{p}+E=0.66-0.0657<p<0.66+0.0657\\ =0.5943<p<0.7257\end{array}$

There is 95% confident that the true proportion of all first- year students at the university who would identify being very well-off as an important personal goal is between 0.5943 and 0.7357.

(b)

To determine

To Explain: that the interval in part (a) provides more information than the test in exercise 52.

Answer to Problem 54E

The confidence interval is not having 73% (or 0.73) which is the national value. Therefore there is sufficient proof to help the claim that the proportion is different (less) than the national value of 73%.

Explanation of Solution

Given:

Result from exercise pat (a):

  $0.5943<p<0.7257$

The confidence interval is not having 73% (or 0.73) which is the national value. Therefore there is sufficient proof to help the claim that the proportion is different (less) than the national value of 73%.