Question

A random sample of $100$of last year’s model of a certain popular car found that $20$had a specific minor defect in the brakes. The automaker adjusted the production process to try to reduce the proportion of cars with the brake problem. A random sample of $350$of this year’s model found that $50$had the minor brake defect.

a. Was the company’s adjustment successful? Carry out an appropriate test to support your answer. b. Based on your conclusion in part (a), which mistake—a Type I error or a Type II error—could have been made? Describe a possible consequence of this error.

Short answer

a. The answer is$P=P(Z>1.39)=P(Z<-1.39)=0.0823$

b. Failure to reject the null hypothesis is a type II mistake.

Step-by-step solution

Part (a) Step 1: Given information

We have to tell about was the company’s adjustment successful

Part (a) Step 2: Explanation

It has been given that

$x_1=20,x_2=50,n_1=100,n_2=350$

${\hat{p}}_2=\frac{x_2}{n_2}=\frac{50}{350}\approx 0.1429$

$z=\frac{{\hat{p}}_1-{\hat{p}}_2}{\sqrt{{\hat{p}}_p\left(1-{\hat{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}=\frac{0.2-0.1429}{\sqrt{0.1556(1-0.1556)}\sqrt{\frac{1}{100}+\frac{1}{350}}}\approx 1.39$

The values using the table

$P=P(Z>1.39)=P(Z<-1.39)=0.0823$

Part (b) Step 1: Given information

We have to tell possible consequence of this error.

Part (b) Step 2: Explanation

The null hypothesis H was not rejected.

• Type I blunder: H should be rejected as a null hypothesis, if H 0H is true as the null hypothesis.

Failure to reject the null hypothesis is a type II mistake.

If the null hypothesis H is false, then

It's only feasible that we made.

• Type II error because we didn't reject the null hypothesis H .