Answer to Question #294624 in Statistics and Probability for Yaku

Question

a) Find the appropriate rejection regions for the large-sample z-test in these cases:

i) A right-tailed test with $\alpha$ = 0.01;

ii) A two-tailed test at 5% signicance level;

iii) A left-tailed test with $\alpha$ = 0.05.

b) Find the p-value for the following large-sample z-tests:

i) A right-tailed test with observed z = 1.15;

ii) A two-tailed test with observed z = -2.78;

iii) A left-tailed test with observed z = -1.81.

c) Suppose that an allergist wishes to test the hypothesis that at least

30% of the public is allergic to some cheese products. Explain how the

allergist could commit:

i) a type I error.

ii) a type II error.

d) A large manufacturing firm is being charged with discrimination in its

hiring practices.

i) What hypothesis is being tested if a jury commits type I error by

finding the firm guilty?

ii) What hypothesis is being tested if a jury commits type II error

by finding the firm guilty?

Answer 1

a) Find the appropriate rejection regions for the large-sample z-test in these cases:

i) A right-tailed test with $\alpha$ = 0.01;

Define p ( z > z₀ ) = 0.01

The rejection region of a right tailed test with  $\alpha$ = 0.01 contains all z scores above the z score z₀ that

has a probability of 0.01 to its right.

Taking p ( z < z₀ ) = 1 - p( z >z₀ ) = 1- 0.01 = 0.99

determining the z score that corresponds to 0.99 from the normal table we get 2.55

Thus z₀ = 2.33.

Hence the rejection region contains all z scores greater than 2.33

ii) A two-tailed test at 5% significance level;

The rejection region of a two tailed test at 5% significance level = 0.05 contains all z scores below the z score - z₀ and above the z score z₀ that has a probability of 0.05/2 = 0.025

Define p ( z < -=z₀ ) = 0.025

Determining the z score that corresponds to 0.025 from the normal table we have -1.96

Thus z₀ = -1.96.

Hence the rejection region contains all z scores smaller than -1.96 and all z scores greater than 1.96

iii) A left-tailed test with α = 0.05.

The rejection region of a left tailed test with  $\alpha$ = 0.05 contains all z scores below the z score z₀ that

has a probability of 0.05 to its left

Define p ( z < z₀ ) = 0.05

Determining the z score that corresponds to 0.05 from the normal table we have -1.645

Thus z₀ = -1.645

Hence the rejection region contains all z scores smaller than -1.645

.

b) Find the p-value for the following large-sample z-tests:

i) A right-tailed test with observed z = 1.15;

p-value = 0.125072

ii) A two-tailed test with observed z = -2.78;

p-value = 0.005436

iii) A left-tailed test with observed z = -1.81.

p-value = 0.035148

c) Suppose that an allergist wishes to test the hypothesis that at least

30% of the public is allergic to some cheese products. Explain how the

allergist could commit:

we first define the null and alternative hypothesis as below

H₀: at least 30% of the public is allergic to some cheese products. ( null hypothesis)

H₁: less than 30% of the public is allergic to some cheese products. ( alternative hypothesis)

i) a type I error.

Reject the null hypothesis although it is true mean that less than 30% of cheese product cause allergy.

ii) a type II error.

At least 30% of people are allergic to cheese products, although they are less than 30%.

d) A large manufacturing firm is being charged with discrimination in its

hiring practices.

i) What hypothesis is being tested if a jury commits type I error by

finding the firm guilty?

The hypothesis will be that the firm is not guilty as type I error occurs during a hypothesis testing when a null hypothesis is rejected ,even though it is accurate and should not be rejected.

ii) What hypothesis is being tested if a jury commits type II error

by finding the firm guilty?

The hypothesis will be the firm is guilty as type II error means not rejecting the null hypothesis when its actually false.