Question

A supplier of Apple Air pods sent a shipment of 10,000 trousers to a retail store. The supplier’s contract with retailer says that shipments must have a defect rate less than .05The retail store looks at 200 sets of Air pods from the shipment and finds that there is some type of defect in 4 of them.

Assume the null hypothesis is H₀: p = .05 and the alternative hypothesis is H₁: p < .05.  Assume the statistic used to test the hypothesis is the sample proportion, and that the significance level chosen is .05.  If the true proportion is .03, what is the probability of failing to reject H₀?

Answer 1

1)

H0: p = .05 v/s H1: p < .05

X = 4
n = 200  
p-hat = X/n = 0.020 = 4/200
po = 0.05

test statistic, z = (phat-p)/sqrt(p*(1-p)/n)
z = (0.02-0.05)/SQRT(0.05*(1-0.05)/200)
z = -1.947

critical value, -z(a) = -z(0.05) = -1.645

Since -z < -z(a), i reject the null hypothesis at 5% level of significance and conclude that p < .05

2)

the probability of failing to reject the null hypothesis is P value.

In other words, the p-value is the probability that the null hypothesis is true.

If the true proportion is 0.03,

p-hat = X/n= 0.030

po= 0.050

test statistic, z = (phat-p)/sqrt(p*(1-p)/n)

=(0.03-0.05)/SQRT(0.05*(1-0.05)/200)

-1.298

p-value

P(Z<z)

P(z<-1.297771)

=NORMSDIST(-1.297771)

0.097183019

If the true proportion is .03, what is the probability of failing to reject H0 is 0.097