Question

The manufacturer of a new racecar engine claims that the proportion p1 p 1 of engine failures due to overheating for this new engine be less than the proportion p2 p 2 of engine failures due to overheating of the old engines. To test this statement, NASCAR took a random sample of 155 of the new racecar engines and 115 of the old engines. They found that 7 of the new racecar engines and 18 of the old engines failed due to overheating during the test. Does NASCAR have enough evidence to reject the manufacturer's claim about the new racecar engine? Use a significance level of α=0.05 α = 0.05 for the test.

Answer 1

For sample 1, we have that the sample size is N1​=155, the number of favorable cases is X1​=7, so then the sample proportion is

For sample 2, we have that the sample size is N2​=115, the number of favorable cases is X2​=18, so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p1​=p2​

Ha:p1​

This corresponds to a left-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a left-tailed test is zc​=−1.64.

The rejection region for this left-tailed test is R={z:z<−1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) The decision about the null hypothesis

Since it is observed that z=−3.122the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0009, and since p=0.0009<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is less than p2​, at the 0.05 significance level. Hence there is sufficient evidence to support the claim about the new racecar engine.

Graphically