Chapter 8.3, Problem 66E

To determine

To calculate: The critical value $t^{*}$ when the confidence level is 90%.

Answer to Problem 66E

The critical value is 1.671, with technology critical value is 1.665 and the advantage for more accurate is that 90% of the confidence interval contains the true population mean.

Explanation of Solution

Given Information:

Sample size $n=77$

Confidence interval $c=90\%$

Formula Used Degree of freedom $df=n-1$

Calculation:

Find the degree of freedom, use the formula $df=n-1$ .

  $\begin{array}{l}df=n-1\\ =77-1\\ =76\end{array}$

Table B does not contain a row with $df=76$ .Use the row that is closest to $df=76$ .

Thus, $df=60$

The confidence level is 90%.

Convert 90% into decimal.

  $\frac{90}{100}=0.90$

Find the value of the column.

  $\begin{array}{c}\frac{1-c}{2}=\frac{1-0.90}{2}\\ =0.05\end{array}$

From table B find critical value $t^{*}$ , use the row and column.

Thus, the critical value $t^{*}\text{is}1.671$ .

Find the critical value, use technology for $df=76$ and $2p\left(X>x\right)=0.01$ .

Thus, critical value $t^{*}\text{is}1.665$

Thus, the critical value is 1.671 and with technology critical value is 1.665.

Higher the accuracy of degree of freedom more accurate will be the confidence interval. Thus, 90% of the confidence interval contains the true population mean for the sample size of 77.

Hence, 90% of the confidence interval contains the true population mean.