Question

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

A manufacturer considers his production process to be out of control when defects exceed 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

A manufacturer considers his production process to be out of control when defects exceed 3%.

So the null hypothesis(H₀ ) and the alternative hypothesis ( H_a ) are as follows:

H₀ : P = 0.03

H_a : P > 0.03

We can used one sample proportion z test.

Let' write given information.

n = sample size = 85

Sample proportion = 0.059

level of significance = 0.01

Formula of test statistic Z is as follows:

Plug the required values in the above formula we get:

So test statisyic value = 1.57

p-value = P( Z > 1.57) = 1 - P(Z <1.57) ........( 1 )

From Z - table , P( Z < 1.57) = 0.9418

Plug this value in equation ( 1 ) , we get:

P-value = 1 - 0.9418 = 0.0582

P-value = 0.103

Decision rule:

1) If p-value < level of significance (alpha) then we reject null hypothesis

2) If p-value > level of significance (alpha) then we fail to reject null hypothesis.

Here p value = 0.0582 > 0.01 so we used 2nd rule.

That is we fail to reject null hypothesis

Conclusion: At 1% level of significance there are not sufficient evidence to say that the sample data indicates the proportion of defectives are greater than 3%.

So we fail to reject the claim of the manufacturer on the basis of the given sample information of the items.

there is not sufficient evidence to warrant rejection of the claim that the mean height of women is equal to 160.4 cm