Q. 65 - The Practice of Statistics for AP

Question

65. Critical value What critical value$t*$ from Table $B$would you use for a $99\%$ confidence interval for the population mean based on an SRS of size $58$? If possible, use technology to find a more accurate value of $t*$. What advantage does the more accurate $df$provide?

Step-by-Step Solution

Answer

The critical value is $2.678$, then the critical value with technology is $2.665$, with the advantage of being more exact because $99\%$of the confidence interval includes the population mean.

Step 1: Given information

To determine the critical value$t*$ from Table $B$ with $99\%$confidence interval for the population mean.

Step 2: Explanation

The sample size was reduced by $1$ degree of freedom.

Where, sample size n is $58$.
$$\begin{gathered} df=n-1 \\ =58-1 \\ =57 \end{gathered}$$
Because there is no row with$df=57$ in table $B,$ use the row with the smallest degrees of freedom that is closest to $df=57$:
$df=50$
Table $B$contains the critical value $t^{*}$in the row with $df=50$and the column with $(1-c)/2=0.005$:
$t^{*}=2.678$

Step 3: Explanation

A less accurate degrees of freedom will lead to significant in the confidence interval being more than $99\%$ confident of containing the true population mean as the critical value is larger. Using technology ( sample like Student's t-Distribution calculator) with$df=57$ and $2P(X>x)=0.01$, then users receive $x=2.665$, and therefore the more accurate critical value is $t*=2.665$.
The benefit of more accurate degrees of freedom is that the confidence interval will be more accurate as well, so the $99\%$ confidence interval will be $99\%$confident of containing the true population mean of an SRS of size $58$.

Therefore, $99\%$ of the confidence interval contains the true population mean.