Question

A teacher's union is on strike for higher wages. The union claims that the mean salary for teachers is at most $45,000 per year, but a legislator thinks that it is higher than $45,000 per year. The legislator does not want to reject the union's claim, however, unless the evidence is very strong against it. Because of this, a 1% significance level is chosen. In order to test the union's claim, a random sample of 100 teachers is obtained, and the sample salary mean X̄ = $46,200. And assume that the population deviation σ = $4,000. A. Conduct a test of significance and provide a conclusion of your test—Please follow the steps in detail (e.g., describing the hypotheses, etc). B. What is the type I error in this example? And explain what this type I error implies. C. Suppose a car manufacturer claims that when a certain model is driven at a speed of 50 miles per hour on a test track, the mileage follows a normal distribution with mean 30 miles per gallon and standard deviation 4 miles per gallon. A consumer advocate thinks that the manufacturer is overestimating average mileage. The advocate decides to test the null hypothesis H0: μ = 30 against the one-sided alternative hypothesis H1: μ < 30. Suppose the consumer advocate tests a random sample of n = 25 cars and uses a 5% level of significance. Find the power of the test if the true mean is μ = 28.5. D. The attention span of children (ages 3 to 5) is claimed to be normally distributed with a mean of 15 minutes and a standard deviation of 4 minutes. A test is to be performed to decide if the average attention span of these kids is really this short or if it is longer. You decide to test the hypotheses H0: μ0 = 15 versus Ha: μ > 15 at the 5% level of significance. A sample of 10 children will watch a TV show they have never seen before, and the time until they walk away from the show will be recorded. If, in fact, the true mean attention span of these kids is 18 minutes, what is the probability of a Type II error?

A teacher's union is on strike for higher wages. The union claims that the mean salary for teachers is at most $45,000 per year, but a legislator thinks that it is higher than $45,000 per year. The legislator does not want to reject the union's claim, however, unless the evidence is very strong against it. Because of this, a 1% significance level is chosen. In order to test the union's claim, a random sample of 100 teachers is obtained, and the sample salary mean X̄ = $46,200. And assume that the population deviation σ = $4,000.

A. Conduct a test of significance and provide a conclusion of your test—Please follow the steps in detail (e.g., describing the hypotheses, etc).

B. What is the type I error in this example? And explain what this type I error implies.

C. Suppose a car manufacturer claims that when a certain model is driven at a speed of 50 miles per hour on a test track, the mileage follows a normal distribution with mean 30 miles per gallon and standard deviation 4 miles per gallon. A consumer advocate thinks that the manufacturer is overestimating average mileage. The advocate decides to test the null hypothesis H0: μ = 30 against the one-sided alternative hypothesis H1: μ < 30. Suppose the consumer advocate tests a random sample of n = 25 cars and uses a 5% level of significance. Find the power of the test if the true mean is μ = 28.5.

D. The attention span of children (ages 3 to 5) is claimed to be normally distributed with a mean of 15 minutes and a standard deviation of 4 minutes. A test is to be performed to decide if the average attention span of these kids is really this short or if it is longer. You decide to test the hypotheses H0: μ0 = 15 versus Ha: μ > 15 at the 5% level of significance. A sample of 10 children will watch a TV show they have never seen before, and the time until they walk away from the show will be recorded. If, in fact, the true mean attention span of these kids is 18 minutes, what is the probability of a Type II error?

Added by Raul B.

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