Let ? denote the true average tread life of a certain type of tire. Consider testing H0: ? = 30,000 versus Ha: ? > 30,000 based on a sample of size n = 16 from a normal population distribution with ? = 1500. A test with ? = 0.01 requires z? = z0.01 = 2.33. The probability of making a type II error when ? = 31,000 is? = Φ 2.33 + 30,000 − 31,0001500/ 16 = Φ (rounded to two decimal places) = 0.3669Since z0.1 = 1.28, the requirement that the level 0.01 test also have ?(31,000) = 0.1 necessitatesn = 1500(2.33 + 1.28)30,000 − 31,000 2 = (−5.42)2 = (rounded to two decimal places).The sample size must be an integer, so n = (rounded up to the next whole number) tires should be used.

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Answered: Let ? denote the true average tread…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

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Let ? denote the true average tread life of a certain type of tire. Consider testing H₀: ? = 30,000 versus H_a: ? > 30,000 based on a sample of size n = 16 from a normal population distribution with ? = 1500. A test with ? = 0.01 requires z_? = z_0.01 = 2.33. The probability of making a type II error when ? = 31,000 is

= Φ

2.33 +

30,000 − 31,000

1500/

= Φ

(rounded to two decimal places)

= 0.3669

Since z_0.1 = 1.28, the requirement that the level 0.01 test also have ?(31,000) = 0.1 necessitates

n =

1500(2.33 + 1.28)

30,000 − 31,000

	2

= (−5.42)² =

(rounded to two decimal places).

The sample size must be an integer, so n = (rounded up to the next whole number) tires should be used.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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