Question

A technician compares repair costs for two types of microwave ovens (type I and type II). He believes that the repair cost for type I ovens is greater than the repair cost for type II ovens. A sample of 35 type I ovens has a mean repair cost of $78.81. The population standard deviation for the repair of type I ovens is known to be $13.96. A sample of 31 type II ovens has a mean repair cost of $76.47. The population standard deviation for the repair of type II ovens is known to be $12.45. Conduct a hypothesis test of the technician's claim at the 0.05 level of significance. Let μ1 be the true mean repair cost for type I ovens and μ2 be the true mean repair cost for type II ovens.

Step 1 of 5: State the null and alternative hypotheses for the test.
Step 2 of 5: Compute the value of the test statistic. Round your answer to two decimal places.

Step 3 of 5: Find the p-value associated with the test statistic. Round your answer to four decimal places.

Step 4 of 5: Make the decision for the hypothesis test.

Step 5 of 5: State the conclusion of the hypothesis test. (Sufficient evidence or not sufficient evidence)

Answer 1

Since , the population standard deviations are known.

Therefore , use normal distribution.

Step 1 of 5 : The null and alternative hypothesis is ,

$H_0:\mu_1=\mu_2$

$H_a:\mu_1>\mu_2$

The test is right tailed test.

Step 2 of 5 : The value of the test statistic is ,

$Z_{stat}=\frac{\bar{X_1}-\bar{X_2}}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}=\frac{78.81-76.47}{\sqrt{\frac{13.96^2}{35}+\frac{12.45^2}{31}}}=0.72$

Step 3 of 5 : The p-value is ,

p-value=$P(Z>|Z_{stat}|)=P(Z>0.72)=1-P(Z\leq 0.72)=1-\Phi(0.72)$

$=1-0.7642=0.2358$ ; From standard normal distribution table

Step 4 of 5 : Decision : Here , p-value>0.05 significance level

Therefore , fail to reject the null hypothesis.

Step 5 of 5 : Conclusion : There is not sufficient evidence to to support the technician's claim.