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?

Short 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-by-step solution

Given Information

Given

$n=77$

$c=90\%=0.90$

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

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

Explanation

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

$df=60$

The critical value $t^{*}$can be found in table $\text{B}$in the row with $df=50$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 $df=77$and $2P(X>x)=0.1$, then you obtain $x=1.665$and thus a more accurate critical value is $t^{*}=1.665$.

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