9. Hypothesis Testing for One Sample

If we do not reject the null hypothesis when the statement in the alternative hypothesis is true, we have made a Type ________ error.

In Problems 7–12, the null and alternative hypotheses are given. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. What parameter is being tested?

H₀: σ = 7.8

H₁: σ ≠ 7.8

In Problems 21–32, state the conclusion based on the results of the test.  

For the hypotheses in Problem 19, the null hypothesis is rejected.

MigrainesAccording to the Centers for Disease Control, 15.2% of American adults experience migraine headaches. Stress is a major contributor to the frequency and intensity of headaches. A massage therapist feels that she has a technique that can reduce the frequency and intensity of migraine headaches.

c. Suppose, in fact, that the percentage of patients in the program who experience migraine headaches is 15.3%. Was a Type I or Type II error committed?

The ________ ___ ___________ is the probability of making a Type I error. 

Explain what a P-value is. What is the criterion for rejecting the null hypothesis using the P-value approach?

a. Determine the critical value for a right-tailed test of a population standard deviation with 18 degrees of freedom at the α = 0.05 level of significance.

b. Determine the critical value for a left-tailed test of a population standard deviation for a sample of size n = 23 at the α = 0.1 level of significance.

c. Determine the critical values for a two-tailed test of a population standard deviation for a sample of size n = 30 at the α = 0.05 level of significance.

Waiting in LineOne aspect of queuing theory is to consider waiting time in lines. A fast-food chain is trying to determine whether it should switch from having four cash registers with four separate lines to four cash registers with a single line. It has been determined that the mean wait-time in both lines is equal. However, the chain is uncertain about which line has less variability in wait time. From experience, the chain knows that the wait times in the four separate lines are normally distributed with σ = 1.2 minutes. In a study, the chain reconfigured five restaurants to have a single line and measured the wait times for 50 randomly selected customers. The sample standard deviation was determined to be s = 0.84 minute. Is the variability in wait time less for a single line than for multiple lines at the α = 0.05 level of significance?

A test is conducted at the alpha = 0.05 level of significance. What is the probability of a Type I error?

Explain the procedure for testing a hypothesis using the P-value Approach. What is the criterion for judging whether to reject the null hypothesis?

Drive-thru Time (See Problem 21 in Section 10.3.) The mean waiting time at the drive-thru of a fast-food restaurant from the time an order is placed to the time the order is received is 84.3 seconds. A manager devises a new drive-thru system that he believes will decrease wait time. He initiates the new system at his restaurant and measures the wait time for 10 randomly selected orders. The wait times are provided below.

a. The goal of the research is to determine if wait times have decreased as a result of the new system. Based on this research objective, state the null and alternative hypotheses.

According to the National Center for Health Statistics, 22.4% of adults are smokers. A random sample of 300 adults is obtained.

b. In a random sample of 300 adults, what is the probability that at least 50 are smokers?

Explain the difference between “accepting” and “not rejecting” a null hypothesis.