Question

The mean height of 36 randomly selected New York City students is 68.2 inches, with a standard deviation of 4.18 inches. Finish the hypothesis test of the claim that the mean height of all New York City students is greater than 67 inches. Use a significance level of a= 0.05

We ________ have sufficient evidence to __________ the claim.

Answer 1

The provided sample mean is 68.2 and the known population standard deviation is σ=4.18, and the sample size is n = 36

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ≤67

Ha: μ>67

This corresponds to a right-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=0.05, and the critical value for a right-tailed test is z_c = 1.64

(3) Test Statistics

The z-statistic is computed as follows:

$z = \frac{\bar X - \mu_0}{\sigma/\sqrt{n}} = \frac{ 68.2 - 67}{ 4.18/\sqrt{ 36}} = 1.722$

(4) Decision about the null hypothesis

Since it is observed that z = 1.722 >z_c​=1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0425 , and since p = 0.0425 <0.05, 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 greater than 67, at the 0.05 significance level.

We _______do_ have sufficient evidence to _____accept_____ the claim.