Question

Question 3
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.

(a) State the null and alternative hypotheses for the test. [3 marks]
(b) Calculate the value of the test statistic for this test. [2 marks]
(c) Determine the critical region(s) for this test. [2 marks]
(d) State the conclusion of this test. Give a reason for your answer. [3 marks]
Total 10 marks

Answer 1

a) H₀ : p = 0.03 vs H₁: p > 0.03

H_a represents the inspector’s claim.
So this is a right -- tailed test.
n = 85
pˆ = 0.059
α = 0.01 p₀ =0.03
b) : Test Statistic for Proportion
z= pˆ−p_{0 /}(p₀⋅(1−p₀)/ n )^0.5

z =(0.059 −0.03)/ {(0.03×0.97) /85}^0.5

z=1.57
c) Find the P-value

P-value = the area to the right of the Test Statistic = 1 - 0.9418 (obtained from Table)
= 0.0582
d) : Conclusion: Compare the P-value with Significance level α
Since P-value=0.0582>0.05=α ,

fail to Reject H0 .

That is: There is not sufficient evidence to support the
inspector's claim that production is out of control.