BASIC PRACTICE OF STATISTICS+LAUNCHPAD
BASIC PRACTICE OF STATISTICS+LAUNCHPAD
8th Edition
ISBN: 9781319053093
Author: Moore
Publisher: MAC HIGHER
Question
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Chapter 18, Problem 18.52E

(a)

To determine

To find: The probability of a Type I error.

(a)

Expert Solution
Check Mark

Answer to Problem 18.52E

The probability of a Type I error is 0.5.

Explanation of Solution

Calculation:

The probability of Type I error:

Formula for probability of Type I error is,

P(Type I error)=P(x¯>0)|μ=0=P(x¯μ(σn)>0μ(σn))|μ=0=P(Z>00(σn))

                                 =P(Z>0)=1P(Z0)

From Table A: Standard Normal Cumulative Proportions, the value of P(Z0) is 0.5.

Therefore,

P(Type I error)=1P(Z0)=10.5=0.5

Thus, the probability that Type I error is 0.5.

(b)

To determine

To find: The probability of a Type II error when μ=0.5 .

(b)

Expert Solution
Check Mark

Answer to Problem 18.52E

The probability of a Type II error when μ=0.5 is 0.1587.

Explanation of Solution

Calculation:

Probability of Type II error:

P(Type II error)=P(x¯0)|μ=0.5=P(x¯μ(σn)0μ(σn))|μ=0.5=P(Z00.5(2.525))=P(Z1)

From Table A: Standard Normal Cumulative Proportions, the value of P(Z1) is 0.1587.

Therefore,

P(Type II error)=0.1587

Thus, the probability of a Type II error when μ=0.5 is 0.1587.

(c)

To determine

To find: The probability of a Type II error when μ=1.0 .

(c)

Expert Solution
Check Mark

Answer to Problem 18.52E

The probability of a Type II error when μ=1.0 is 0.0228.

Explanation of Solution

Calculation:

Probability of Type II error:

P(Type II error)=P(x¯0)|μ=1.0=P(x¯μ(σn)0μ(σn))|μ=1.0=P(Z01.0(2.525))=P(Z2)

From Table A: Standard Normal Cumulative Proportions, the value of P(Z2) is 0.0228.

Therefore,

P(Type II error)=0.0228

Thus, the probability of a Type II error when μ=1.0 is 0.0228.

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