Exam 14: Exercises

Which of the following will increase the value of the power in a statistical test of hypotheses?

A researcher wishes to determine if aerobic exercise improves mental performance immediately following the exercise. He plans to have high school students participate in 30 minutes of aerobic exercise and then take a standard test of their reasoning skills. Suppose the scores of high school students on this test of reasoning skills immediately after 30 minutes of aerobic exercise follow a Normal distribution, with mean μ and standard deviation σ= 4. Suppose also that, in the general population of all high school students, scores on the test of reasoning skills follow a Normal distribution, with mean 25 and standard deviation σ= 4. The researcher, therefore, decides to test the following hypotheses: H₀: μ= 25, H_a: μ> 25 To do so, the researcher has 10,000 high school students perform 30 minutes of aerobic exercise and then, immediately following the exercise, take the test. The mean score for these students is x̄= 25.2 and the P-value is less than 0.0001. Suppose that two high school students decide to see if they get the same results as the researcher. They both do 30 minutes of aerobic exercise and, immediately following the exercise, take the test. The mean of their scores is = 25.2 seconds, the same as that of the researcher. It is appropriate to conclude which of the following?

The makers of the pomegranate juice POM Wonderful claimed in advertisements that the juice helps with erectile dysfunction; this claim was based on a study showing a significant improvement after drinking POM Wonderful. However, the Federal Trade Commission (FTC) filed a formal complaint "alleging that the erectile dysfunction claims are false and unsubstantiated because the study on which the company relied did not show that POM Juice was any more effective than a placebo." What can we say about the FTC complaint?

The level of nitrogen oxides (NO_x) in the exhaust of cars of a particular model varies Normally, with standard deviation σ= 0.05 gram per mile (g/mi). Government regulations call for NO_x emissions no higher than 0.3 g/mi. In testing H₀: μ= 0.3 versus H_a: μ< 0.3 using significance level 0.05, we find that the power of this test against an alternative of μ= 0.28 based on a random sample of size n = 40 is 0.808. What impact would using a larger sample size (n = 50) have?

The distribution of total body protein in adult men with liver cirrhosis is approximately Normal, with standard deviation σ= 0.1 kg. Which sample size, n, would be required to obtain a margin of error of 0.02 kg for a 90% confidence interval for the mean total body protein of adult men with liver cirrhosis?

Your advisor has asked you to reduce the margin of error on your confidence interval from ±5 mg/l to ±1 mg/l. The population standard deviation is known to be 8 mg/. What should you do?

The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels more than 200 mg/dl is Normally distributed, with mean (in milligrams) μ and standard deviation σ= 3. The brand advertises that eating its oatmeal for breakfast daily for one month will produce a mean decrease in cholesterol of more than 10 points for people with cholesterol levels more than 200 mg/dl, but you believe that the mean decrease in cholesterol is actually less than advertised. To explore this question, you test the following hypotheses at the 0.05 level: H₀: μ= 10, H_a: μ< 10 You obtain a P-value of 0.052. Which of the following is TRUE?

The distribution of total body protein in healthy adult men is approximately Normal, with standard deviation σ= 0.1 kg. A 95% confidence interval for the true mean total body protein of adult men with liver cirrhosis, based on 58 randomly selected adult males with liver cirrhosis, has a margin of error of 0.026 kg. If the study had enrolled fewer subjects, how would the resulting margin of error for a 95% confidence interval compare?

You plan to construct a confidence interval for the mean μ of a Normal population with (known) standard deviation σ. Which of the following actions will reduce the size of the margin of error?

Which of the following describe good practice when assessing the validity of any test of hypotheses?

The z procedures and many other procedures designed for Normal distributions rely on Normality of the sampling distribution of x̄.

People tend to give more socially acceptable answers when asked a sensitive question in person rather than anonymously via computer. When computing a 95% confidence interval for a population parameter, what does the margin of error cover?

When performing a test of hypotheses, what is a Type II error?

Which of the following describes good practice when assessing the validity of any test of hypotheses?

The distribution of total body protein in healthy adult men is approximately Normal, with standard deviation σ= 0.1 kg. A 95% confidence interval for the true mean total body protein of adult men with liver cirrhosis, based on 58 randomly selected adult males with liver cirrhosis, has a margin of error of 0.026 kg. How would a 90% confidence interval based on the same study compare to the 95% confidence interval?

The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels more than 200 mg/dl is Normally distributed, with mean (in milligrams) μ and standard deviation σ= 3. The brand advertises that eating its oatmeal for breakfast daily for one month will produce a mean decrease in cholesterol of more than 10 points for people with cholesterol levels more than 200 mg/dl, but you believe that the mean decrease in cholesterol is actually less than advertised. To explore this question, you test the following hypotheses at the 0.05 level: H₀: μ= 10, H_a: μ< 10 You carry out the level 0.05 test by randomly selecting 100 people with cholesterol levels more than 200 mg/dl and, after they have eaten this brand of oatmeal for one month, computing the mean decrease x̄ in the subjects' cholesterol levels. What is the probability of a Type II error for your test at μ= 8.5?

Because large random samples have small chance variation, very small deviations from the null hypothesis can be highly significant if the sample is large.

A researcher plans to conduct a test of hypotheses at the α= 0.05 significance level. She designs her study to have a power of 0.80 at a particular alternative value for the parameter of interest. What is the probability that the researcher will commit a Type II error for the particular alternative value of the parameter at which she computed the power?

The time (in number of days) until maturity of a certain variety of tomato plant is Normally distributed, with mean and standard deviation ?= 2.4. You select a simple random sample of four plants of this variety and measure the time until maturity. The four times, in days, are as follows: 63 $\quad$ 69 $\quad$ 62 $\quad$ 66 If you wanted the margin of error for the 99% confidence interval to be ±1 inch, which size of simple random sample should you select?

A researcher plans to conduct a test of hypotheses at the α= 0.05 significance level. She designs her study to have a power of 0.80 at a particular alternative value of the parameter of interest. What is the probability that the researcher will commit a Type I error?