Statistics and Estimating

1. Suppose a wildlife service wishes to estimate, with 99% confidence, the mean number of days of hunting per hunter for all licensed hunters in the state during a given season, with a bound on the error of estimation equal to 2 hunting days. How many hunters must be included in the survey? Assume that the data collected in earlier surveys have shown o = 10.

2. We wish to estimate the mean serum indirect bilirubin level of 4-day-old infants. The mean for a sample of 160 infants was found to be 5.98 mg/100 cc. Assume that bilirubin levels in 4-day-old infants are approximately normally distributed with a standard deviation of 3.5 mg/100 cc. Construct a 95% confidence interval for the true mean bilirubin level in 4-day-old infants.

3. A researcher studied the effects of pancuronium-induced muscle relaxation on circulating plasma volume on newborn infants weighing more than 1700 grams who required respiratory assistance within the first 24 hours of birth and met other clinical criteria. The following table is a result of measurements of plasma volume (ml) made during mechanical ventilation.

a. You may assume equal standard deviations. Construct a 98% confidence interval for the difference of the two population means.

b. What are your conclusions? Do you think there is a difference between the two population means?

4. Consider the following scenario:
Dr. Jeffrey M. Barrett of Lakeland, Florida, reported data on eight cases of umbilical cord prolapse. The maternal ages were 25, 28, 17, 26, 27, 22, 25, and 30. He was interested in determining if he could conclude that the mean age of the population of his sample was greater than 20 years. Let a = .01. The null and alternative hypotheses for this problem are defined as:
subject group sample size sample mean sample st. dev.
paralyzed 5 48.0 8.1
nonparalyzed 7 56.7 8.1
H0: u = 20 and u > 20.

a. Explain the meaning of a Type I error in the context of this problem.
b. Explain the meaning of a Type II error in the context of this problem.
c. Suppose Dr. Barrett expanded his study to include a total of 50 subjects and then calculated a test statistic of z = 2.33 for his hypothesis test. What is the associated p-value? What is the conclusion for this test?

5. A local weight-loss company suggests that the average client loses 15 pounds during the first month. A consumer advocate group feels that the actual number of pounds lost is much less than this. To test the claim, they select 30 of the clients at random and obtain the following data:

Pounds Lost During the 1st Month: 16 20 14 15 14 18 19 20 9 13 20 11 12 18 13 17 11 21 14 15 15 13 17 12 9 20 16 18 21 11

A hypothesis test is conducted.

a. State the null and alternative hypotheses.
b. The test statistic is t = 0.6107 with p-value = 0.7269.

Based on this information, what is the conclusion of the hypothesis test? Justify your answer.

6. The manufacturer of a particular all-season tire claims that the tires last for 22,000 miles. After purchasing the tires you discover that yours did not last the full 22,000 miles. Suppose that a sample of 100 tires made by that manufacturer lasted on average 21,819 miles with a sample standard deviation of 1,295 miles. Is there sufficient evidence to refute the manufacturer's claim that the tires last 22,000 miles? Let a = 0.05. Assume that the population standard deviation is o = 1300.

a. Define the null and alternative hypotheses.
b. Find the appropriate rejection region.
c. Compute the test statistic.
d. What is your conclusion? Explain.

7. The following data presents tests scores on a Behavioral Vignettes Test for self-help skill teaching for the primary parents of a random sample of families with mentally retarded children before and after a training program.

Before 7 6 10 16 8 13 8 14
After 11 14 16 17 9 15 9 17

Based on this information, can you conclude that the training program is effective? Use a significance level of 0.05. State your assumptions