Question

 Test at the α = .01 significance level whether the mean of a random sample of size n = 16 is statistically significantly less than 10. If the distribution from which the sample was taken is normal, x¯ = 8.4 and σ2 = 10.24. What are the null and alternative hypotheses for this test?

Answer 1

Hypothesized Population Mean $\mu=10$

Population Standard Deviation $\sigma=\sqrt{\sigma^2}=\sqrt{10.24}=3.2$

Sample Size $n=16$

Sample Mean $\bar{x}=8.4$

Significance Level $\alpha=0.01$

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

$H_0: \mu\geq10$

$H_1: \mu<10$

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

Based on the information provided, the significance level is $\alpha=0.01$ and the critical value for  left-tailed test is $z_c=-2.3263.$ 

The rejection region for this left-tailed test is $R=\{z:z<-2.3263\}.$

The $z$ - statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{8.4-10}{3.2/\sqrt{16}}=-2$$

Since it is observed that $z=-2>-2.3263=z_c,$ it is then concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$ is less than $10,$ at the $\alpha=0.01$ significance level.

Using the P-value approach: The p-value for one-tailed, the significance level $\alpha=0.01, z=-2, \text{left-tailed}$ is $p=0.02275,$ and since $p=0.02275>0.01=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$ is less than $10,$ at the $\alpha=0.01$ significance level.