Stats Professor Exam 3

Directions: Try to take the exam as if it were an actual test.  When you finish, click the problems one-by-one to check your answers.  There is a view answer link to just see the text solution, but if you got the problem wrong, you should watch the included video as well.

1. Assume the following sample data is to be used to estimate the population mean.  Find the margin of error:

98% confidence, n = 17, sample mean = 68.0, and s = 2.8.

2. Allstate Insurance claims that the average commute distance is less than 15 miles.  Twenty-six randomly selected commuters are surveyed, and it is found that they drove an average of 14.4 miles during their commute.  Their standard deviation was 7.2 miles.  Test All State’s claim at the 5% significance level.

3. To help consumers assess the risks they are taking, the Food and Drug Administration (FDA) publishes the amount of tar found in all brands of cigarettes.  For a particular brand of cigarette, FDA tests yielded a mean tar level of 1.89 milligrams and standard deviation of 0.4 milligrams for a sample of nine cigarettes. Construct a 99% confidence interval for the mean tar content of this brand of cigarette.

4. Researchers claim that the average amount of lean mass that can be put on by an experienced athlete (> 21 yrs old) over the course of a year without performance enhancing drugs is less than 2 pounds.  A random sample of 23 experienced athletes followed a strict diet that consisted of 40% protein, 40% carbs, and 20% healthy fats.  At the end of one year, the change in lean mass was recorded for each athlete.  The mean change was 0.9 lbs.  The standard deviation was 0.3 lbs.  Express the claim, the null and alternative hypotheses, and find the test statistic that would be used to test the researcher’s claim.

5. A random sample of 2000 voters yielded 530 who reported being in favor of changing the constitution to allow foreign born people to hold the office of President.  A 99% confidence interval was constructed for the true proportion of people who are in favor of the change.  The resulting interval was as follows: [0.240, 0.290].  If an immigrant group claims that the majority of the public supports the change, does this interval contradict their claim?

6. What is the best way to reduce the margin of error in a confidence interval?

7. We have created a 95% confidence interval for μ with the result (148, 196). What conclusion will we make if we test H0: μ = 200 vs. Ha:μ ≠ 200 at α = 5%?

8. Shooting ranges need to know the average amount of time that shooters will typically spend on the range to decide whether to charge per hour or to have a single daily rate for unlimited time on the range.  For this reason, Texas Shooting Range wants to estimate the mean time that shooters will spend on the range per session if they charge a daily rate for unlimited time on the range.  They would like to estimate this mean within 5 minutes and with 98% reliability.  If the range’s initial experiences indicate that the standard deviation for the amount of time spent on the range is 22 minutes, how many shooters must be sampled for the range to get the information it desires?

9. Assume that the data has a normal distribution and the number of observations is fifty. Find the critical z value used to test a null hypothesis, if the significance level is 1% and we are conducting a left-tailed test. 

10. Use the following confidence level and sample data to find the margin of error E.  Exam scores: 99% confidence, n = 84, sample mean 67.9, and σ = 14.8.

11. Assume that the data has a normal distribution and the test statistic is Z = 1.98. Find the p-value used to test the null hypothesis, μ ≤ 170.  If the significance level is 2.5%, what is your initial conclusion?

12. Find the appropriate Za/2 or ta/2 critical value: confidence level = 95% and n = 14.

13. A laboratory tested 83 compact fluorescent bulbs for mercury content and found that the mean amount of mercury was 5.33 milligrams with a standard deviation of 1.1 milligrams. Construct a 95% confidence interval for the true mean mercury content, μ, of all such bulbs

14. A tire manufacturer claims that their tires have a mean lifetime equal to 75,000 miles (assuming regular rotations of the tires are performed).  A sample of 36 of their tires are randomly selected and tested.  They have a mean lifetime 73,125 miles with a standard deviation of 4,800 miles.  Use the p-value method of hypothesis testing to test the company’s claim at the 2% significance level. 

15. A doctor claims that the proportion of Americans exercising the recommended minimum of 45 minutes per day is less than ten percent.  A sample of 550 Americans reveals that 52 of them exercise at least 45 minutes per day.  At the 10% significance level test the doctor’s claim.