Chapter 8: Q 61. (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 $95 %$ confidence interval based on $n = 10$ randomly selected observations
b. $A 99 %$ confidence interval from an SRS of $20$ observations
c. A $90 %$ confidence interval based on a random sample of $77$ individuals

Short Answer

Expert verified

a. The critical value is $2.262$

b. The critical value is $2.861$

c. The critical value is $1.671$

Step by step solution

Part (a) Step 1: Given Information

It is given that Confidence Level $= 95 %$

$n = 10$

Part (a) Step 2: Calculation

Level of significance, $(α) = 1 - 0.95 = 0.05$

Degree of $d f = n - 1$

$= 10 - 1 = 9$

At $5 %$ , level of significance

$t_{α / 2, d f} = z_{0.05 / 2, 9} = 2.262$

The critical value is $2.262$

Part (b) Step 1: Given Information

It is given that Confidence Level $= 99 %$

$n = 20$

Part (b) Step 2: Calculation

Level of Significance, $(α) = 1 - 0.99 = 0.01$

Degree of freedom, $d f = n - 1$

$= 20 - 1 = 19$

At $1 %$ level of significance,

$t_{α / 2, d f} = t_{0.01 / 2, 19} = 2.861$

The critical value is $2.861$

Part (c) Step 5: Given Infromation

It is given that Confidence Level $= 90 %$

$n = 77$

Part (c) Step 2: Calculation

Level of significance, $(α) = 1 - 0.90 = 0.10$

Degree of freedom, $d f = n - 1$

$= 77 - 1 = 76$

At $10 %$ level of significance, critical value is

$t_{α / 2, d f} = t_{0.10 / 2, 76} = 1.671$

The critical value is $1.671$

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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 5: Given Infromation

Part (c) Step 2: Calculation

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