Question

A sample of 10 items provides a sample standard deviation, s of 11. Consider the following hypothesis:

H0: σ2 ≤ 40 Ha: σ2 > 40.

A.Use both the p-value approach and critical value approach to test the above hypothesis at α = .05. [4 points]

B.What is your conclusion? [ 2 point]

C.Construct a 95% confidence interval for the population variance .[ 2 points]

Answer 1

Solution:

Given: Sample size = n = 10 , Sample Standard Deviation = s = 11

Hypothesis:

H₀: σ² ≤ 40 H_a: σ² > 40.

Part A) Use both the p-value approach and critical value approach to test the above hypothesis             at α = .05

Find test statistic:

p-value:

df = n - 1= 10 - 1 = 9

Look in Chi-square critical value table for df = 9 row and find the interval in which Chi-square = 27.225 fall.

Since Chi-square test statistic value = 27.225 > 23.589 , corresponding p-value would be less than 0.005.

Thus p-value < 0.005.

Since p-value < 0.005, that is: p value < 0.05 level of significance, we reject null hypothesis H0.

Critical value approach:

df = 9 and level of significance = 0.05

Chi-square critical value = 16.919

Since Chi-square test statistic value = 27.225 > Chi-square criitical value = 16.919, we reject null hypothesis H0.

Part B) .What is your conclusion?

We conclude that: variance is greater than 40.

Part C.Construct a 95% confidence interval for the population variance .

Formula:

where is Chi-square critical value for right tail area = 0.05 /2 = 0.025

and df = 9

Thus  

and

is Chi-square critical value for left tail area = 0.025

but here we use area to the right side.

If Area to the left tail is 0.025 , then area to right side of 0.025 area is = 1 - 0.025 = 0.975

Thus

Thus a 95% confidence interval for the population variance is .