Question

Determine the given and compute the appropriate test statistic of the problem below. Construct the rejection region of the problem below

A manufacturer of face mask has developed a new face mask design. He claims that the new design has an average profit increase of 10% with a standard deviation of 3%. Test the hypothesis that the new face mask design average profit increase of is not 10% if a random sample of 50 face mask is tested with an average profit increase of 4%. Use 10% level of significance.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=10$

$H_1:\mu\not=10$

This corresponds to a two-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 $\alpha = 0.1,$ and the critical value for a two-tailed test is $z_c = 1.6449.$

The rejection region for this two-tailed test is $R = \{z: |z| > 1.6449\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{4-10}{3/\sqrt{50}}=-14.142$$

Since it is observed that $|z| = 14.142 > 1.6449=z_c ,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is $p=2P(Z<-14.142)=0,$  p and since $p = 0 < 0.1=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$

is different than 10, at the $\alpha = 0.1$ significance level.