 # Show that for any Δ &gt; 0, when the population distribution is normal and σ is known, the two-tailed test satisfies β ( μ 0 − Δ) = β ( μ 0 + Δ), so that β ( μ ′) is symmetric about μ 0 . ### Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809 ### Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809

#### Solutions

Chapter
Section
Chapter 8.2, Problem 27E
Textbook Problem

## Show that for any Δ > 0, when the population distribution is normal and σ is known, the two-tailed test satisfies β(μ0 − Δ) = β(μ0 + Δ), so that β(μ′) is symmetric about μ0.

Expert Solution
To determine

Show that for two tailed test, β(μ0Δ)=β(μ0+Δ).

### Explanation of Solution

Given info:

The population distribution is normal and the population standard deviation is known. The additional information is that if β(μ0Δ)=β(μ0+Δ) then β(μ') will be symmetric about μ0.

Calculation:

From the definition of β, for two tailed test,

β(μ0Δ)=Φ(zα2+Δσn)Φ(zα2+Δσn) (1)

Where Φ() is the cdf of the standard normal distribution.

It is known that Φ(z)=1Φ(z)

Assuming zα2+Δσn as –z, Φ(zα2+Δσn)=1Φ(zα2Δσn) (2)

And assuming zα2+Δσn as –z, Φ(zα2+Δσn)=1Φ(zα2Δσ<

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