Question

Consider the following hypothesis test. H0: μ ≤ 50 Ha: μ > 50

A sample of 60 is used and the population standard deviation is 8. Use the critical value approach to state your conclusion for each of the following sample results. Use α = 0.05. (Round your answers to two decimal places.)

(a) x = 52.3

Find the value of the test statistic. =

State the critical values for the rejection rule. (If the test is one-tailed, enter NONE for the unused tail.)

test statistic ≤ =

test statistic ≥ =

State your conclusion.

a. Do not reject H0. There is sufficient evidence to conclude that μ > 50.

b. Reject H0. There is sufficient evidence to conclude that μ > 50.

c. Reject H0. There is insufficient evidence to conclude that μ > 50.

d. Do not reject H0. There is insufficient evidence to conclude that μ > 50.

(b) x = 51

Find the value of the test statistic. =

State the critical values for the rejection rule. (If the test is one-tailed, enter NONE for the unused tail.)

test statistic ≤ =

test statistic ≥ =

State your conclusion.

a. Do not reject H0. There is sufficient evidence to conclude that μ > 50.

b. Reject H0. There is sufficient evidence to conclude that μ > 50.

c. Reject H0. There is insufficient evidence to conclude that μ > 50.

d. Do not reject H0. There is insufficient evidence to conclude that μ > 50.

(c) x = 51.8

Find the value of the test statistic. =

State the critical values for the rejection rule. (If the test is one-tailed, enter NONE for the unused tail.)

test statistic ≤ =

test statistic ≥ =

State your conclusion.

a. Do not reject H0. There is sufficient evidence to conclude that μ > 50.

b.Reject H0. There is sufficient evidence to conclude that μ > 50.

c.Reject H0. There is insufficient evidence to conclude that μ > 50.

d. Do not reject H0. There is insufficient evidence to conclude that μ > 50.

Answer 1

Solution:

Part a

Here, we have to use one sample z test for population mean.

H₀: µ ≤ 50 versus H_a: µ > 50

This is an upper tailed or right tailed (one tailed) test.

We are given

Level of significance = α = 0.05

n = 60

σ = 8

Xbar = 52.3

Test statistic formula is given as below:

Z = (Xbar - µ) / [σ/sqrt(n)]

Z = (52.3 – 50) / [8/sqrt(60)]

Z = 2.2270

Test statistic = 2.23

Upper critical value = 1.6449

(by using z-table)

Rejection rule:

Reject H₀ when test statistic ≥ 1.6449

State your conclusion.

Here, Test statistic = 2.23 > Upper critical value = 1.6449

So, we reject the null hypothesis

b. Reject H0. There is sufficient evidence to conclude that μ > 50.

Part b

Here, we have to use one sample z test for population mean.

H₀: µ ≤ 50 versus H_a: µ > 50

This is an upper tailed or right tailed (one tailed) test.

We are given

Level of significance = α = 0.05

n = 60

σ = 8

Xbar = 51

Test statistic formula is given as below:

Z = (Xbar - µ) / [σ/sqrt(n)]

Z = (51 – 50) / [8/sqrt(60)]

Z = 0.9682

Test statistic = 0.97

Upper critical value = 1.6449

(by using z-table)

Rejection rule:

Reject H₀ when test statistic ≥ 1.6449

State your conclusion.

Here, Test statistic = 0.97 < Upper critical value = 1.6449

So, we do not reject the null hypothesis

d. Do not reject H0. There is insufficient evidence to conclude that μ > 50.

Part c

Here, we have to use one sample z test for population mean.

H₀: µ ≤ 50 versus H_a: µ > 50

This is an upper tailed or right tailed (one tailed) test.

We are given

Level of significance = α = 0.05

n = 60

σ = 8

Xbar = 51.8

Test statistic formula is given as below:

Z = (Xbar - µ) / [σ/sqrt(n)]

Z = (51.8 – 50) / [8/sqrt(60)]

Z = 1.7428

Test statistic = 1.74

Upper critical value = 1.6449

(by using z-table)

Rejection rule:

Reject H₀ when test statistic ≥ 1.6449

State your conclusion.

Here, Test statistic = 1.74 > Upper critical value = 1.6449

So, we reject the null hypothesis

b. Reject H0. There is sufficient evidence to conclude that μ > 50.