Question

Consider the following hypotheses:

H₀: μ ≤ 42.2
H_A: μ > 42.2

A sample of 25 observations yields a sample mean of 43.2. Assume that the sample is drawn from a normal population with a population standard deviation of 6.0.

a-1. Find the p-value.

• p-value < 0.01

• 0.01  p-value < 0.025
• 0.025  p-value < 0.05
• 0.05  p-value < 0.10
• p-value  0.10

a-2. What is the conclusion if α = 0.10?
  

• Reject H₀ since the p-value is greater than α.

• Reject H₀ since the p-value is smaller than α.

• Do not reject H₀ since the p-value is greater than α.

• Do not reject H₀ since the p-value is smaller than α.

a-3. Interpret the results at α = 0.10.

• We conclude that the sample mean is greater than 42.2.

• We cannot conclude that the sample mean is greater than 42.2.

• We conclude that the population mean is greater than 42.2.

• We cannot conclude that the population mean is greater than 42.2.

a-4. Calculate the p-value if the above sample mean was based on a sample of 113 observations.

• 0.025  p-value < 0.05
• 0.05  p-value < 0.10
• p-value  0.10

• p-value < 0.01

• 0.01  p-value < 0.025

a-5. What is the conclusion if α = 0.10?
  

• Reject H₀ since the p-value is smaller than α.

• Reject H₀ since the p-value is greater than α.

• Do not reject H₀ since the p-value is smaller than α.

• Do not reject H₀ since the p-value is greater than α.

a-6. Interpret the results at α = 0.10.

• We conclude that the sample mean is greater than 42.2.

• We cannot conclude that the sample mean is greater than 42.2.

• We conclude that the population mean is greater than 42.2.

• We cannot conclude that the population mean is greater than 42.2.

Answer 1

Solution :

The null and alternative hypotheses are as follows :

H₀: μ ≤ 42.2
H_A: μ > 42.2

a-1) To test the hypothesis we shall use z-test. The test statistic is given as follows :

Where, x̅ 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, x̅ = 43.2, σ = 6.0, n = 25 and μ = 42.2

The value of the test statistic is 0.8333.

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 > 0.8333)

P-value = 0.2023

The p-value is greater than 0.10.

a-2) Do not reject H₀ since the p-value is greater than α.

a-3) We cannot conclude that the population mean is greater than 42.2.

a-4) To test the hypothesis we shall use z-test. The test statistic is given as follows :

Where, x̅ 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, x̅ = 43.2, σ = 6.0, n = 113 and μ = 42.2

The value of the test statistic is 1.7717.

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 > 1.7717)

P-value = 0.0382

0.025 < p-value < 0.05

a-5) Reject H₀ since the p-value is smaller than α.

a-6) We conclude that the population mean is greater than 42.2.

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