Question

A certain manufactured product is supposed to contain23% potassium by weight. A sample of 10 specimensof this product had an average percentage of 23.2 witha standard deviation of 0.2. If the mean percentage isfound to differ from 23, the manufacturing processwill be recalibrated.
a. State the appropriate null and alternate hypotheses.
b. Compute the P-value.
c. Should the process be recalibrated? Explain.

Answer 1

Answer:

a. The null and alternative hypothesis are

$H_0: \pi=23\\\\H_1: \pi\neq23$

b. P-value = 0.7181

c. The process should not be recalibrated. We have no enough statistical proof with this sample that the mean is not 23% (the null hypothesis π=23% could not be rejected).

Step-by-step explanation:

We want to know if there is enough statistical evidence to claim that the mean of the process is not 23%.

Then, the null and alternative hypothesis are

$H_0: \pi=23\\\\H_1: \pi\neq23$

We assume a significance level of 0.05.

The sample of size N=10 has a p=0.232.

The standard deviation to is

$\sigma=\sqrt{\frac{\pi(1-\pi)}{N} } =\sqrt{\frac{0.23(1-0.23)}{10} }=0.133$

The test statistic z is given by

$z=\frac{p-\pi-0.5/N}{\sigma}=\frac{0.232-0.2-0.5/10}{0.133}=\frac{-0.048}{0.133}=-0.361$

The two-tailed P-value for z=-0.361 is P=0.7181. This value is bigger than the significance level, so the effect is not significant. The null hypothesis can not be rejected.

That means we have no statistical proof with this sample that the mean is not 23%. The process should not be recalibrated.