A Test for the Variance of the Normal Distribution

Question 1

What is the null and alternative hypothesis in a left-tailed variance test?

Answer 1

H₀: σ² ≥ σ₀², H₁: σ² < σ₀²

Question 2

What is the test statistic for testing variance of a normal distribution?

Answer 2

T = (n−1)S² / σ², which follows a χ² distribution with (n−1) degrees of freedom.

Question 3

What is the rejection region for a left-tailed test of variance?

Answer 3

Reject H₀ if S² < c, where c is based on the critical value from the χ² distribution.

Question 4

How is the critical value c computed in a left-tailed test?

Answer 4

c = (σ₀² / (n−1)) * χ²₁₋α,(n−1)

Question 5

What distribution does (n−1)S² / σ² follow under H₀?

Answer 5

χ² distribution with (n−1) degrees of freedom.

Question 6

How do you calculate sample variance S²?

Answer 6

S² = (1 / (n−1)) * Σ(Xᵢ − X̄)²

Question 7

What R command is used to calculate sample variance?

Answer 7

var(x) or ((sum(x^2)-(sum(x)^2)/n)/(n-1))

Question 8

In the nozzle example, what was the conclusion at α = 0.04?

Answer 8

Fail to reject H₀. No sufficient evidence that σ² > 0.01.

Question 9

What is the rejection region for a right-tailed test of variance?

Answer 9

Reject H₀ if S² > c, where c = (σ₀² / (n−1)) * χ²α,(n−1)

Question 10

Why do we maximize P(S² < c) over σ² ≥ σ₀² in left-tailed test?

Answer 10

Because P(S² < c) decreases as σ² increases, the maximum occurs at σ² = σ₀².