Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access)
Loose-leaf Version for The Basic Practice of Statistics 7e & LaunchPad (Twelve Month Access)
7th Edition
ISBN: 9781319019334
Author: David S. Moore, William I. Notz, Michael A. Fligner
Publisher: W. H. Freeman
Question
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Chapter 18, Problem 18.52E

a.

To determine

The probability of type I error that the test reject H0 when μ=0 .

a.

Expert Solution
Check Mark

Answer to Problem 18.52E

The probability of type I error is 0.5.

Explanation of Solution

Given info:

The hypothesis is given as:

The null hypothesis is,

H0:μ=0

The research hypothesis is,

Ha:μ>0

Simple random sample of size n is 25 and standard deviation is σ=1 .

Calculation:

Type I error:

The probability of rejecting null hypothesis when it is true.

Type I error states that rejecting H0 when it is true and it is denoted by α .

P(x¯>0whenμ=0)=P(x¯μσn>0μσn) when μ=0=P(z>0)=0.5P(z<0)=0.50

Further solve the above equation.

P(x¯>0whenμ=0)=0.5

The 0.5 probability value is obtained by using area under normal curve table.

Thus, the probability of type I error is 0.5.

b.

To determine

The probability of type II error that the test reject H0 when μ=0.25 .

b.

Expert Solution
Check Mark

Answer to Problem 18.52E

The probability of type II error is 0.1056.

Explanation of Solution

Calculation:

Type II error:

The probability of accepting null hypothesis when it is false.

Type II error states that accepting H0 when it is false and it is denoted by β .

P(x¯>0whenμ=0.25)=P(x¯μσn0μσnwhenμ=0.25)=P(z1.25)=P(z>1.25) (Bysymmetric)=0.5P(z<1.25)

Further salve the above equation.

P(x¯>0whenμ=0.25)=0.50.3944=0.1056

The 0.3944 probability value is obtained by using area under normal curve table.

Thus, the probability of type II error is 0.1056.

c.

To determine

The probability of type II error when μ=0.5 .

c.

Expert Solution
Check Mark

Answer to Problem 18.52E

The probability of type II error is 0.0202.

Explanation of Solution

Calculation:

Type II error:

The probability of accepting null hypothesis when it is false.

Type II error states that accepting H0 when it is false and it is denoted by β .

P(x¯>0whenμ=0.5)=P(x¯μσn0μσnwhenμ=0.5)=P(z2.5)=P(z>2.5)                                 (Bysymmetric)=0.5P(Z<2.5)

The 0.4798 probability value is obtained by using area under normal curve table.

Thus,

P(x¯>0whenμ=0.5)=0.50.4798=0.0202

Thus, the probability of type II error is 0.0202.

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