1. A manufacturer of car batteries claims that the batteries will last, on average, three years with a standard deviation of one year. If 5 of these batteries have lifetimes of 1.9, 2.4, 3.0, 3.5 and 4.2 years, construct a 90% confidence interval for and decide if the manufacturers claim that = 1 is valid. Assume the population of battery lives to be normally distributed.

2. Two different brands of latex paint are being considered for use. Fifteen specimens of each type of paint were selected, and the drying times, in hours, were as follows. Paint A : 3.5,2.7,3.9,4.2,3.6,2.7,3.3,5.2,4.2,2.9,4.4,5.2,4.0,4.1,3.4 Paint B : 4.7,3.9,4.5,5.5,4.0,5.3,4.3,6.0,5.2,3.7,5.5,6.2,5.1,5.4,4.8 Find a 99% confidence interval for the difference of population mean dying times for the two paints assuming the populations are normally distributed with equal variance.

3. Two levels (low and high) of insulin doses are given to two groups of diabetic rats to check the insulin-binding capacity, yielding the following summary statistics. Low dose: n1 = 8 x1 = 1.98 s1 = 0.51 High dose: n2 = 13 x2 = 1.30 s2 = 0.35 Assume the populations are normal with unequal variances. Give a 95% confidence interval for the difference in the true mean insulin-binding capacity between the two samples.

4. Those teams who win Formula 1 championships have pit crews who change tires as fast as possible. The following table contain the times (in seconds) that the pit crews of two different teams spent changing tires in 10 randomly selested races per team. Team 1 : 5.613,6.130,5.422,5.947,5.514,5.322,5.690,5.243,5.920,5.859 Team 2 : 5.934,5.335,5.826,4.821,5.664,5.292,5.257,6.245,5.981,5.197

(a) Assuming the samples are from normal populations, compute a 90% confidence interval for the variance ratio 21/22? Looking at the confidence interval can we assume the variances are equal?

(b) Use the result in part (a) to compute a 90% confidence interval for the difference of the population means 1 2. Write your conclusions based on the analysis.

5. A group of engineers working with physicians in a research hospital is developing a new device to measure blood glucose levels. Based on measurements taken from patients in a previous study, the physicians assert that the new device provides blood glucose levels slightly higher than those provided by the old device. To corroborate their suspicion, 15 diabetic patients were randomly selected, and their blood glucose levels were measured with both the new and the old devices. The measurements, in mg /100 ml, are in "glucoselevels.csv" file.

(a) Are the samples independent? Why or why not? (

b) If the blood glucose level is normally distributed random variable, compute a 95% confidence interval for the difference of the population means.

(c) Use the results in (b) to decide whether or not the two devices give the same results.

6. A clinical trial was conducted to determine if a certain type of inoculation has an effect on the incidence of a certain disease. A sample of 1000 rats was kept in a controlled environment for a period of 1 year, and 500 of the rats were given the inoculation. In the group not inoculated, there were 120 incidences of the disease, while 98 of the rats in the inoculated group contracted it. If p1 is the probability of incidence of the disease in inoculated rats and p2 the probability of incidence in inoculated rats, compute a 90% confidence interval for p1 p2.

7. A manufacturer turns out a product item that is labeled either "defective" or "not defective." In order to estimate the proportion defective, a random sample of 100 items is taken from production, and 10 are found to be defective. Following implementation of a quality improvement program, the experiment is conducted again. A new sample of 100 ie taken, and this time only 6 are found to be defective.

(a) Give a 98% confidence interval for p1p2, where p1 is the population proportion defective before improvement and p2 is the proportion defective after improvement.

(b) Is there information in the confidence interval found in (a) that would suggest that p1 >p2? Explain.

8. The number of alpha particles emitted per second by a radio active substance observed by a Geiger counter are as follows. 6,5,8,8,13,11,7,8,7,10,8,4,12,3,11,9,15,12,6,3 A possible model for the data is the Poisson distribution with parameter . Suppose the prior distribution for is a gamma distribution with parameters = 2 and = 3. Find the posterior distribution of given the data and find a 95% credible interval for .