Exam 11: Introduction to Hypothesis Testing

During the last energy crisis, a government official claimed that the average car owner refills the tank when there is more than 3 gallons left. To check the claim, 10 cars were surveyed as they entered a gas station. The amount of gas remaining before refill was measured and recorded as follows (in gallons): 3, 5, 3, 2, 3, 3, 2, 6, 4, and 1. Assume that the amount of gas remaining in tanks is normally distributed with a standard deviation of 1 gallon. Compute the probability of a Type II error and the power of the test if the true average amount of gas remaining in tanks is 3.5 gallons.

A social researcher claims that the average adult listens to the radio less than 26 hours per week. He collects data on 25 individuals' radio listening habits and finds that the mean number of hours that the 25 people spent listening to the radio was 22.4 hours. If the population standard deviation is known to be eight hours, can we conclude at the 1% significance level that he is right?

The p-value of a test is the smallest $\alpha$ at which the null hypothesis can be rejected.

The power of a test is denoted by:

NARRBEGIN: Watching the News Watching the News A researcher claims viewers spend an average of 40 minutes per day watching the news. You think the average is higher than that. In testing your hypotheses H₀: $\mu$ = 40 vs. H₁: $\mu$ > 40, the following information came from your random sample of viewers: $\bar { X }$ = 42 minutes, n = 25. Assume $\sigma$ = 5.5, and $\alpha$ = 0.10.NARREND -{Watching the News Narrative} Calculate the value of the test statistic.

A Type II error is defined as:

Think about a situation where you have a test for a virus. First, you are tested positive or negative. Second, you either really do have the virus or you don't. a.If you actually have the virus but the test did not catch it, which error has been made and what is the impact of that error? b.If you actually don't have the virus but the test says you did, which error is being made and what is the impact of this error? c.Which error is the worst one to commit in this situation and why?

NARRBEGIN: Watching the News Watching the News A researcher claims viewers spend an average of 40 minutes per day watching the news. You think the average is higher than that. In testing your hypotheses H₀: $\mu$ = 40 vs. H₁: $\mu$ > 40, the following information came from your random sample of viewers: $\bar { X }$ = 42 minutes, n = 25. Assume $\sigma$ = 5.5, and $\alpha$ = 0.10.NARREND -{Watching the News Narrative} Set up the rejection region.

In testing the hypotheses H₀: $\mu$ = 75 vs. H₁: $\mu$ < 75, if the value of the test statistic z equals -2.42, then the p-value is:

A Type II error is represented by $\alpha$ ; it is the probability of rejecting a true null hypothesis.

In a criminal trial, a Type II error is made when:

You cannot commit a(n) ____________________ error when the null hypothesis is false.

We cannot commit a Type I error when the:

The p-value of a test is the probability of observing a test statistic at least as extreme as the one computed given that the null hypothesis is true.

If a researcher fails to reject a false null hypothesis he has made a(n) ____________________ error.

A p-value is usually set at 0.05.

Which of the following statements is false regarding the operating characteristic (OC) curve?

Which of the following is an appropriate null hypothesis?

If a hypothesis is rejected at the 0.025 level of significance, it:

Suppose a pickup and delivery company states that their packages arrive within two days or less on average. You want to find out whether the actual average delivery time is longer than this. You conduct a hypothesis test. a.Set up the null and alternative hypotheses. b.Suppose you conclude wrongly that the company's statement about average delivery time is within two days. What type of error is being committed and what is the impact of that error? c.Suppose you conclude wrongly that the delivery company's average time to delivery is in fact longer than two days. What type of error did you commit and what is the impact of this error? d.Which error is worse from the company's standpoint, a Type I or a Type II error? Why? e.Which error is worse from a consumer standpoint, a Type I or a Type II error? Why?