Question

7) An auto manufacturer claims that only 10% of its cars have defective parts. A consumer agency thinks that this claim is too low and randomly tests 100 cars. If 16 cars have defective parts does this indicate, at a 1% level of significance, that the auto manufacturer"s claim is correct?

a) find the critical z-score(s) or t-score(s)

b) compute the test statistic

c) state whether or not to reject the null hypothesis

d) answer the question posed at the end of each problem

Answer 1

The null and alternative hypothesis for this test are-

Null hypothesis, ,(Auto manufacturer claim) i.e., the true proportion of all cars that have defective parts is 0.10 or 10%.

Alternative hypothesis, , i.e., the true proportion of all cars that have defective parts is GREATER THAN 0.10 or 10%.

So we have given the information of randomly selected 100 cars and out of which 16 cars have defective parts. So at significance level of we need to test this hypothesis and make conclusion about the manufacturer's claim.

_____________________________

(a) The test which is used to test this hypothesis is one proportion Z test, then the critical value is given as-

___________________________

(b) The formula for calculating the test statistic is

where,

, the null-hypothesized value for the population parameter

, sample proportion of cars with defective parts.

, sample size

So the test statistic is calculated as

______________________________

(c) Since we are testing aright-tail hypothesis then the decision rule is given as -

Reject null hypothesis if,

The value of test-statistic we have calculated as and the critical value is

Since,

So at significance level of the sample data provides insufficient evidence to reject null hypothesis , hence we fail to reject null hypothesis

_____________________________

(d) At significance level of the sample data provides insufficient evidence to reject null hypothesis , hence, "We fail to reject null hypothesis "

In other words, we can say that sicne we fail to reject the null hypothesis , so we conclude that," the auto manfacturer's claim is correct, that only 10% of its cars have defective parts."