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Expert-verified Found in: Page 518 ### The Practice of Statistics for AP

Book edition 4th
Author(s) David Moore,Daren Starnes,Dan Yates
Pages 809 pages
ISBN 9781319113339 # 57. Critical values What critical value t* from Table B would you use for a confidence interval for the population mean in each of the following situations?(a) A $95%$ confidence interval based on  $n=10$ observations.(b) A $99%$ confidence interval from an SRS of $20$ observations.

(a) The critical value for 95% confidence interval based on n=10 observations is ${t}^{*}=2.262$.

(b)The critical value for 99% confidence interval based an SRS of 20 observations is ${t}^{*}=2.861$.

See the step by step solution

## Part (a) Step 1: Given information

When the confidence level is $95%$ and the population mean is $10$, the critical value $t*$ is calculated.

## Part (a) Step 2: Explanation

Utilize the formula $df=n-1$ to estimate the degree of freedom. Where, the population mean is $n=10$.
$df=n-1$$=10-1=9$The row in table $B$is represented by the degree of freedom $df$.

Convert the confidence level $95%$ into decimal.$\frac{95}{100}=0.95$Determine the column:$\frac{1-c}{2}=\frac{1-0.95}{2}\phantom{\rule{0ex}{0ex}}=0.025$Using the table$B$, to determine the critical value $t*,$ for row $9$ and the column $0.025$:${t}^{*}=2.262$

Therefore, the critical value is ${t}^{*}=2.262$.

## Part (b) Step 1: Given information

When the confidence level is $99%$ and the population mean is $20$, the critical value $t$ is calculated.

## Part (b) Step 2: Explanation

Using the formula $df=n-1$, to determine the degree of freedom. Where, the population mean is $n=20$.
$df=n-1=20-1=19$.

Convert the confidence level $99%$ into decimal as:

$\frac{99}{100}=0.99$

Determine the column by:

$\frac{1-c}{2}=\frac{1-0.99}{2}\phantom{\rule{0ex}{0ex}}=0.005$

Using the table B, determine the critical value ${t}^{*}$, for row $19$ and the column $0.005$:

${t}^{*}=2.861$

Therefore, the critical value is ${t}^{*}=2.861$. ### Want to see more solutions like these? 