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Q. 57

Expert-verified

The Practice of Statistics for AP

Found in: Page 518

Book edition 4th

Author(s) David Moore,Daren Starnes,Dan Yates

Pages 809 pages

ISBN 9781319113339

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Short Answer

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 $t^{*} = 2.861$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

Convert the confidence level $95 %$ into decimal. $\frac{95}{100} = 0.95$ Determine the column: $\frac{1 - c}{2} = \frac{1 - 0.95}{2} = 0.025$ 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 $d f = n - 1$ , to determine the degree of freedom. Where, the population mean is $n = 20$ .
$d f = n - 1 = 20 - 1 = 19$ .

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

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

Determine the column by:

$\frac{1 - c}{2} = \frac{1 - 0.99}{2} = 0.005$

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 $t^{*} = 2.861$ .

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