Question

An Izod impact test was performed on 20 specimens of PVC pipe. The obtained standard deviation was s = 0.328. Assume that the random sample was drawn form a normal population N (µ, σ^2 ), with unknown mean value µ and unknown variance σ^2 .

a) Give a 99% confidence interval for the variance σ^2 .

b) Give a 99% upper confidence bound for the variance σ^2 .

c) Test the hypothesis

H0 : σ^2 = 0.1,

against the two-sided alternative

H1 : σ^2 ≠ 0.1,

at the significance level α = 0.01.

d) Explain the connection between your two-sided confidence interval and the significance test.

Answer 1

a) Give a 99% confidence interval for the variance σ^2 .

99% CI for σ using Chi-Square

(0.230, 0.547)

b) Give a 99% upper confidence bound for the variance σ^2 .

Upper confidence bound = 0.547

c) Test the hypothesis

H0 : σ^2 = 0.1,

against the two-sided alternative

H1 : σ^2 ≠ 0.1,

at the significance level α = 0.01.

The hypothesis being tested is:

Null hypothesis	H₀: σ² = 0.1
Alternative hypothesis	H₁ : σ² ≠ 0.1

Method	Test Statistic	DF	P-Value
Chi-Square	20.44	19	0.737

The p-value is 0.737.

Since the p-value (0.737) is greater than the significance level (0.01), we cannot reject the null hypothesis.

Therefore, we have insufficient evidence to conclude that σ² = 0.1.

d) Explain the connection between your two-sided confidence interval and the significance test.

The confidence interval does not contain 0.1 and from the hypothesis test, we have insufficient evidence to conclude that σ² = 0.1. Therefore, both methods gives the same answer.