Question

A manufacturer considers his production process to be out of control when defects exceed 4) 3%. In a random sample of 85 items, the defect rate is 5.9% but the manager claims that this is only a sample fluctuation and production is not really out of control. At the 0.01 level of significance, test the manager's claim.

Answer 1

Answer:

If, p-value<α reject the null hypothesis, hence the production process is not really out of control.  

Step-by-step explanation:

From the information, observe that a manufacturer considers his production process to be out of control when defects exceeds 3%.  Consider a random sample of 85 items, the defect rate is 5.9%.

Here, The claim is that this is the only sample fluctuation and production is not really out of control.

Consider a null and alternative hypothesis:

Null hypothesis, H_o: The production process is not out of control when the defect does not exceed 3%.

that is H_o: p<=0.03

Alternative hypothesis, H_a: the production process to be out of control when defect exceeds 3%

That is H_a: p>0.03:

Level of significance α= 0.01

Test statistics under null hypothesis. $z= \frac{p-p}{\sqrt{\frac{pq}{n} } }$

$z= \frac{0.059-0.03}{\sqrt{\frac{0.03(1-0.03)}{85} } }$

=$\frac{0.029}{0.0185}$

= 1.57

Calculative p-values as follows

P(z>1.57)= 1-P(z<=1.57)

= 1-0.9418

=0.0582

compare the p value with level significance

If, p-value<α reject the null hypothesis, hence the production process is not really out of control.