9. Hypothesis Testing for One Sample

c. Determine the critical values for a two-tailed test of a population mean at the α = 0.01 level of significance based on a sample size of n = 33.

b. Determine the critical value for a left-tailed test of a population mean at the α = 0.01 level of significance based on a sample size of n = 40.

You Explain It! ESP Suppose an acquaintance claims to have the ability to determine the birth month of randomly selected individuals. To test such a claim, you randomly select 80 individuals and ask the acquaintance to state the birth month of the individual. If the individual has the ability to determine birth month, then the proportion of correct birth months should exceed 1/12, the rate one would expect from simply guessing.

a. State the null and alternative hypotheses for this experiment.

Interesting Results Suppose you wish to find out the answer to the age-old question, "Do Americans prefer Coke or Pepsi?" You conduct a blind taste test in which individuals are randomly asked to drink one of the colas first, followed by the other cola, and then asked to disclose which drink they prefer. Results of your taste test indicate that 53 of 100 individuals prefer Pepsi.

c. Suppose you changed the level of significance in conducting the hypothesis test to α = 0.01. What would happen to the range of values of p₀ for which the null hypothesis is not rejected? Why does this make sense?

Efficacy of e-Cigs Do electronic cigarettes assist in helping individuals quit smoking? Researchers found 300 current smokers to volunteer for a study in which each was randomly assigned to one of three treatment groups. Group 1 received an electronic cigarette (e-cig) in which each cartridge contained 7.2 mg of nicotine, Group 2 received an e-cig that contained 5.4 mg of nicotine, and Group 3 received an e-cig that contained no nicotine. The subjects did not know which group they were assigned. During the course of the 52-week intervention, subjects dropped out of the study. At the end of the study 65 subjects remained in Group 1, 63 in Group 2, and 55 in Group 3. After 52 weeks, it was determined via questionnaire whether the subject quit smoking entirely. Results of the study are presented in the following table.

g. Write a conclusion for this hypothesis test.

If we reject the null hypothesis when the statement in the null hypothesis is true, we have made a Type ________ error.

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.