Critical Values of P

Question

2. The prospective purchaser of several new shipments of waterproof gloves hopes they are as good as the old shipments, which had a 10% rate of defective pairs. But he fears that they may be worse. So for each shipment, he takes a random sample of 100 pairs and counts the proportion P of defective pairs so that he can run a classical test. The level of this test was determined by such relevant factors as the cost of a bad shipment, the cost of an alternative supplier, and the new shipper’s reputation. When all these factors were taken into account, suppose the appropriate level of this test was aplha = 0.09.

a. State the null and alternative hypotheses, in words and symbols.

b. What is the critical value of P? That is, how large must P be in order to reject the null hypothesis (i.e. reject the shipment)

Answer 1

This purchaser is comparing the quality rate of new shipments to the previous rate of 10%, and wants to find out if it is worse. Worse in this case means a higher defect rate, so the claim would be that p > 10%.

a) The null hypothesis is that the rate has not changed, and the alternative hypothesis is that the rate has gotten worse.

H₀: p = 0.10

H_A: p > 0.10

 

b) This is a right tailed test, since the alternative hypothesis is >. Look in the body of a z-chart for 1 - α = 1- 0.09 = 0.91 or use the inverse normal function on your calculator or computer to find that z_α ≈ 1.34.

 

The mean for your distribution will be the proportion from the null hypothesis, p = 0.10.

 

The standard deviation is √(p(1-p)/n) = √(0.10*0.90/100) = 0.03.

 

Your critical value is then P = 0.10 + 1.34(0.03) = 0.1402