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?

Short answer

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

The technology used for$t^{*}=1.665$

Step-by-step solution

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 $t^{*}$.

Explanation

The given value is written as

$n=77$

$c=90\%=0.90$

The sample size was reduced by one degree of freedom

$df=n-1=77-1=76$

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 $2P(X>x)=0.1$

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).

$Table:t^{*}=1.671$

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