Question

Consider the following hypothesis test.

H0: μ ≤ 25

Ha: μ > 25

A sample of 40 provided a sample mean of 26.6. The population standard deviation is 6.

(a) Find the value of the test statistic. (Round your answer to two decimal places.)

(b) Find the p-value. (Round your answer to four decimal places.)

1) p-value =

(c) At  α = 0.01, state your conclusion.

1) Reject H₀. There is sufficient evidence to conclude that μ > 25.

2) Reject H₀. There is insufficient evidence to conclude that μ > 25.     

3) Do not reject H₀. There is sufficient evidence to conclude that μ > 25.

4) Do not reject H₀. There is insufficient evidence to conclude that μ > 25.

(d) State the critical values for the rejection rule. (Round your answer to two decimal places. If the test is one-tailed, enter NONE for the unused tail.)

1) test statistic≤ _______?

2) test statistic≥ _______?

State your conclusion.

1) Reject H₀. There is sufficient evidence to conclude that μ > 25.

2) Reject H₀. There is insufficient evidence to conclude that μ > 25.     

3) Do not reject H₀. There is sufficient evidence to conclude that μ > 25.

4) Do not reject H₀. There is insufficient evidence to conclude that μ > 25.

Answer 1

Solution :

The null and alternative hypotheses are as follows :

a) To test the hypothesis we shall use z-test for single mean. The test statistic is given as follows :

Where, x̄ is sample mean, σ is population standard deviation, n is sample size and μ is hypothesized value of population mean under H​​​​​​0.

We have, x̄ = 26.6, σ = 6, n = 40 and  μ = 25

On rounding to two decimal places we get, Z = 1.69.

The value of the test statistic is 1.69.

b) Since, our test is right-tailed test, therefore we shall obtain right-tailed p-value for the test statistic. The right-tailed p-value is given as follows :

p-value = P(Z > value of the test statistic)

p-value = P(Z > 1.6865)

p-value = 0.0458

The p-value is 0.0458.

c) Significance level (α) = 0.01

p-value = 0.0458

(0.0458 > 0.01)

Since, p-value is greater than the significance level of 0.01, therefore we shall be fail to reject H​​​​​​0 at α = 0.01.

At α = 0.01, There is insufficient evidence to conclude that μ > 25.

Do not reject H0. There is insufficient evidence to conclude that μ > 25.

Hence, 4th option is correct.

d) The test is one-tailed (right-tailed). The one-tailed critical value at α = 0.01 is 2.33.

1) test statistic ≤ none

2) test statistic ≥ 2.33

Since, test statistic is not falling in critical region, therefore we shall be fail to reject H0.

Do not reject H0. There is insufficient evidence to conclude that μ > 25.

Hence, 4th option is correct.

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