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 200 of the new racecar engines and 180 of the old engines. They found that 24 of the new racecar engines and 14 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=0.1 for the test.

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

Step 2 of 6: Find the vales of the two sample proportions P1 and P2.Round your answers three decimal places.

Step 3 of 6: Compute the weighted estimate p. p(line above). Round your answers 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 that,

n₁ = 200

x₁ = 180

n₂ = 24

x₂ = 14

Level of significance = = 0.1

Step 1 of 6 :

This a right(one)-tailed test.

The null and alternative hypothesis is,

Ho: p₁ p₂

Ha: p₁ > p₂

Step 2 of 6:

Point estimate = sample proportion = ₁ = x₁ / n₁ = 0.900

Point estimate = sample proportion = ₂ = x₂ / n₂ = 0.583

Step 3 of 6:

The value of the pooled proportion is computed as,

= ( x₁ + x₂ ) / ( n₁ + n₂ )

= (180 + 14 ) / ( 200 + 24)

= 0.866

1 - = 0.134

Step 4 of 6:

Test statistics

z = (₁ - ₂ ) / *(1-) ( 1/n₁ + 1/n₂ )

= (0.900 - 0.583 ) / (0.866 * 0.134 ) (1/200 + 1/24)

= 4.30

Step 5 of 6:

P-value = P(Z > z )

= 1 - P(Z < 4.30 )

= 1 - 1

= 0.0000

Step 6 of 6:

The p-value is p = 0.0, and since p = 0. < 0.1, it is concluded that the null hypothesis is rejected.