Q. 66 - The Practice of Statistics for AP

Question

Critical value What critical value $t^{*}$ 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 . What advantage does the more accurate df provide? 

Step-by-Step Solution

Answer

The accurate value of $t^{*}=1.671$

The technology used for $t^{*}=1.665$

Step 1: Given information

Critical value $t^{*}$ from table B would you use for a $90\%$ confidence

SRS has the size $77$

Find the accurate value and technology of $t^{*}$.

Step 2: Explanation

The given value is written as 

$n=77$

$c=90\%=0.90$

The sample size was reduced by one degree of freedom

$d f=n-1=77-1=76$

Because there is no row with $df=76$ in table B, use the row with the smallest degrees of freedom that is closest to $df=76$

$df=60$

The critical value of $t^{*}$is found in table B with $df=50$

which has the column $(1-c) / 2=0.05$

$t^{*}=1.671$

When we include the technology $df=77$ and $2 P(X>x)=0.1$

Therefore we get $x=1.665$ which is the more accurate critical value of $t^{*}.$

Because more accurate degrees of freedom mean more accurate confidence intervals, the $90\%$ confidence interval will be $90\%$ confident of containing the true population mean of an 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).

$Table: t^{*}=1.671$

$Techno\log y: t^{*}=1.665$.