Question

The manufacturer of a new racecar engine claims that the proportion of engine failures due to overheating for this new engine, (p1), will be no higher than the proportion of engine failures due to overheating of the old engines, (p2). To test this statement, NASCAR took a random sample of 225 of the new racecar engines and 215 of the old engines. They found that 27 of the new racecar engines and 16 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 for the test.

Step 1 of 6 : State the null and alternative hypotheses for the test. ep 1 of 6 : State the null and alternative hypotheses for the test

Step 2 of 6 : Find the values of the two sample proportions, pˆ1and pˆ2. Round your answers to three decimal places

Step 3 of 6 : Compute the weighted estimate of p, _p. Round your answer to three decimal places

Step 4 of 6 : Compute the value of the test statistic. Round your answer to two decimal places

Step 5 of 6 : Find the P-value for the hypothesis test. Round your answer to four decimal places. Step 6 of 6 : Make the decision to reject or fail to reject the null hypothesis

Answer 1

Solution:

Given:

Claim: the proportion of engine failures due to overheating for this new engine, (p₁), will be no higher than the proportion of engine failures due to overheating of the old engines, (p₂).

The new race car engines:

n₁ = 225

x₁ = 27

The old engines:

n₂ = 215

x₂ = 16

Level of significance = α = 0.05

Step 1 of 6 : State the null and alternative hypotheses for the test.

Vs

Step 2 of 6 : Find the values of the two sample proportions, pˆ₁and pˆ₂.

and

Step 3 of 6 : Compute the weighted estimate of p.

Step 4 of 6 : Compute the value of the test statistic.

Step 5 of 6 : Find the P-value for the hypothesis test.

For right tailed test , P-value is:

P-value = P(Z > z test statistic)

P-value = P(Z > 1.61 )

P-value = 1 - P(Z < 1.61 )

Look in z table for z = 1.6 and 0.01 and find corresponding area.

P( Z < 1.61) = 0.9463

thus

P-value = 1 - P(Z < 1.61 )

P-value = 1 -0.9463

P-value =0.0537

Step 6 of 6 : Make the decision to reject or fail to reject the null hypothesis

Decision Rule:
Reject null hypothesis H₀, if P-value < 0.05 level of significance, otherwise we fail to reject H₀

Since P-value =0.0537 > 0.05 level of significance, we fail to reject the null hypothesis H₀.