Answer to Question #217257 in Statistics and Probability for Chaeng

Question

Bamboo craft shop is engaged in making sofa sets made from indigenous materials and exported to other countries. Records show that the weekly production of sofa set follows a normal distribution with a mean of 20 and a standard deviation of 2. The shop owner had undergone series of seminars and after undergoing such seminars, she introduces new methods and techniques in the procedure of sofa making and increased the number of her workers. During, the last 52 weeks, she observed a weekly production of 23 sofa sets. At the 0.05 significance level, can the shop owner conclude that there is a change in the weekly production of sofa set?

Answer 1

The following null and alternative hypotheses need to be tested:

$\mu=20$

$\mu\not=20$

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 $\alpha=0.05,$ and the critical value for a two-tailed test is $z_c=1.96.$

The rejection region for this two-tailed test is $R=\{z:|z|>1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{23-20}{2/\sqrt{52}}=10.817$$

Since it is observed that $|z|=10.817>1.96=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is $p=2P(Z>10.817)=0,$ and since $p=0<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is different than $20,$ at the $\alpha=0.05$ significance level.

Therefore, there is enough evidence to claim that there is a change in the weekly production of sofa set, at the $\alpha=0.05$ significance level.