Answer:
a. The null and alternative hypothesis are
[tex]H_0: \pi=23\\\\H_1: \pi\neq23[/tex]
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
[tex]H_0: \pi=23\\\\H_1: \pi\neq23[/tex]
We assume a significance level of 0.05.
The sample of size N=10 has a p=0.232.
The standard deviation to is
[tex]\sigma=\sqrt{\frac{\pi(1-\pi)}{N} } =\sqrt{\frac{0.23(1-0.23)}{10} }=0.133[/tex]
The test statistic z is given by
[tex]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[/tex]
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.
