Chapter 8: Q 62. (page 541)

Critical values What critical value $t *$ from Table B should be used for a confidence interval for the population mean in each of the following situations?
a. A $90 %$ confidence interval based on $n = 12$ randomly selected observations
b. A $95 %$ confidence interval from an SRS of $30$ observations
c. A $99 %$ confidence interval based on a random sample of size $58$

Short Answer

Expert verified

a. The critical value is $1.796$ .

b. The critical value is $2.045$ .

c. The critical value is $2.678$ .

Step by step solution

Part (a) Step 1: Given Information

It is given that $n = 12$

Confidence Level $= 0.90$

Part (a) Step 2: Calculation

Degree of freedom is calculated as:

$d f = n - 1 = 12 - 1 = 11$

Level of Significance, $a = 0.10$

EXCEL formula $= TINV (0.1, 11)$ gives

$t_{11, a / 2} = 1.796$

The critical value is $1.796$

Part (b) Step 1: Given Information

It is given that $n = 30$

Confidence Level $= 0.95$

Part (b) Step 2: Calculation

Degree of freedom is calculated as:

$d f = n - 1 = 30 - 1 = 29$

Level of significance, $a = 0.05$

EXCEL formula $= TINV (0.05, 29)$ gives

$t_{11, a / 2} = 1.045$

The critical value is $1.045$

Part (c) Step 1: Given Information

It is given that $n = 58$

Confidence Level $0.999$

Part (c) Step 2: Calculation

Degree of freedom is $d f = n - 1 = 58 - 1 = 57$

Level of Significance is $a = 0.01$

EXCEL formula $= TINV (0.01, 57)$ gives

$t_{11, a / 2} = 2.678$

The critical value is $2.678$

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Short Answer

Step by step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Calculation

Part (b) Step 1: Given Information

Part (b) Step 2: Calculation

Part (c) Step 1: Given Information

Part (c) Step 2: Calculation

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