Hypothesis Test for one variance

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Section 11.5 Hypothesis Test for one variance

Proceeding in a similar manner, we can also perform hypothesis testing on variances using the

χ^{2}

-distribution to determine probabilities.

H_{0} : σ^{2} = σ_{0}^{2} H_{a} : σ^{2} \neq σ_{0}^{2}

or

H_{0} : σ^{2} \geq σ_{0}^{2} H_{a} : σ^{2} < σ_{0}^{2}

or

H_{0} : σ^{2} \leq σ_{0}^{2} H_{a} : σ^{2} > σ_{0}^{2}

test statistic = T = (n - 1) \frac{s^{2}}{σ_{0}^{2}}

For two-tailed, reject if

T > χ_{1 - α / 2, n - 1}^{2} or T < χ_{α / 2, n - 1}^{2}

and for one-tailed to the right if

T > χ_{1 - α, n - 1}^{2}

and for one-tailed to the left if

T < χ_{α, n - 1}^{2} .

Technically, for the two-tailed test you could pick T-values so that the total probability sums to

α

in any fashion but generally this probability is split evenly between the two tails as noted above.

Checkpoint 11.5.1. WebWorK - Confidence Interval for $σ$ .

Use a

α = 0.01

significance level to test the claim that

σ = 17

if the sample statistics include

n = 11,

\overset{―}{x} = 106,

and

s = 23.

The test statistic is

The smaller critical number is

The bigger critical number is

What is your conclusion?

There is not sufficient evidence to warrant the rejection of the claim that the population standard deviation is equal to 17
There is sufficient evidence to warrant the rejection of the claim that the population standard deviation is equal to 17

18.3044982698962

2.15586

25.1882

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