Q. 66 - The Practice of Statistics for AP

Question

What critical value $t^{\star }$ from Table B would you use for a $90\%$ confidence interval for the population mean based on an SRS of size $77$? If possible, use technology to find a more accurate value of $t^{\star }$. What advantage does the more accurate df provide?

Step-by-Step Solution

Answer

The advantage of the more accurate degrees of freedom is that the confidence interval is going to be more accurate as well, thus the $90\%$ confidence interval will be $90\%$ confident of containing the true population mean of a SRS of size $77$(while the less accurate degrees of freedom will result in the confidence interval being more than $90\%$ confident of containing the true population mean as the critical value is larger).

Step 1: Given Information

Given

$n=77$

$c=90\%=0.90$

The degrees of freedom is the sample size decreased by $1$:

$$\begin{gathered} d f=n-1 \\ =77-1 \\ =76 \end{gathered}$$

Step 2: Explanation

Since table $\mathrm{B}$ does not contain a row with $d f=76$, use the row with a smaller degree of freedom that is closest to $d f=76$ :

$d f=60$

The critical value $t^{*}$ can be found in table $\text{B}$ in the row with $d f=50$ and in the column with $(1-c) / 2=0.05$ :$t^{*}=1.671$

When using technology (like for example, the Student's t-Distribution calculator on https://homepage.stat.uiowa.edu/$~$mbognar/applets/t.html) with $d f=77$ and $2 P(X>x)=0.1$, then you obtain $x=1.665$ and thus a more accurate critical value is $t^{*}=1.665$.

$$\begin{gathered} Table: t*=1.671 \\ Techno\log y: t*=1.665 \end{gathered}$$