Question

Consider the following hypotheses:

H₀: μ ≥ 180
H_A: μ < 180

A sample of 83 observations results in a sample mean of 175. The population standard deviation is known to be 21. (You may find it useful to reference the appropriate table: z table or t table)


a-1. Calculate the value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
  

a-2. Find the p-value.
  

• p-value 0.10

• p-value < 0.01

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

b. Does the above sample evidence enable us to reject the null hypothesis at α = 0.10?
  

• Yes since the p-value is less than the significance level.

• No since the p-value is greater than the significance level.

• No since the p-value is less than the significance level.

• Yes since the p-value is greater than the significance level.

c. Does the above sample evidence enable us to reject the null hypothesis at α = 0.05?
  

• Yes since the p-value is less than the significance level.

• Yes since the p-value is greater than the significance level.

• No since the p-value is less than the significance level.

• No since the p-value is greater than the significance level.

d. Interpret the results at α = 0.05.

• We conclude that the population mean is less than 180.

• We cannot conclude that the population mean is less than 180.

• We conclude that the population proportion differs from 180.

• We conclude that the population proportion equals 180.

Answer 1

The provided sample mean is 175 and the known population standard deviation is σ=21, and the sample size is n = 83

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ≥180

Ha: μ<180

This corresponds to a left-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.1, and the critical value for a left-tailed test is z_c = -1.28

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z = -2.169< z_c ​=−1.28, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.015 , and since p = 0.015 <0.1, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is less than 180, at the 0.1 significance level.

a-1. Calculate the value of the test statistic.

z = -2.169

a-2. Find the p-value.

0.01 p-value < 0.025

b. Does the above sample evidence enable us to reject the null hypothesis at α = 0.10?

Yes since the p-value is less than the significance level.

c. Does the above sample evidence enable us to reject the null hypothesis at α = 0.05?

Yes since the p-value is less than the significance level.

d. Interpret the results at α = 0.05.

We conclude that the population mean is less than 180.