MATHEMATICS DISCIPLINE

UNIVERSITY OF MINNESOTA, MORRIS

MORRIS, MN 56267

1. The age of all college students follow a distribution that is skewed to the right with a mean of 26 years and a standard deviation of 4 years. Find the probability that the mean age for a random sample of 36 students would be between 25 and 27 years.

2. Suppose that there are only five students in an advanced statistics class and the midterm scores of these five students are

70, 78, 80, 80, 95

All possible samples of size 3 that can be selected from this population without replacement and their respective means are given in the following table

a. Construct the sampling distribution of the sample mean

b. Show that the sample mean is an unbiased estimate of the population mean.

3. One of the most pressing problems in high-technology industries is computer security. Computer security is typically achieved by use of a password. The problem is that persistent hackers can create programs that enter millions of combinations of symbols into a target system until the correct password is found. The newest systems solve this problem by requiring authorized users to identify themselves by unique body characteristics. For example, a system developed by Palmguard, Inc., tests the hypothesis

H₀: The proposed user is authorized

versus

H_a: The proposed user is unauthorized

by checking characteristics of the proposed user’s palm against those stored in the authorized users’ data bank.

a. Define a Type I error and Type II error for this test. Which is the more serious error? Why?

b. Another successful security system, the EyeDentifyer, "spots authorized computer users by reading the one-of-a-kind patterns formed by the network of minute blood vessels across the retina at the back of the eye." The following table gives the probabilities for the Type I and Type II errors for the Palmguard and EyeDentifyer:

Discuss the differences between these two methods by referring to the above probabilities (interpret each one of these four probabilities). In which one of these methods it is harder to for a hacker to get into the system?

4. A consumer agency wants to estimate the proportion of all drivers who wear seat belts while driving. How large should sample size be so that the agency will be 99% sure the estimate is within 0.03 of the true proportion.

5. In 1996, 32% of the U.S. households owned a personal computer (Fortune, March 18, 1996). In a recent sample of 850 U.S. households, 305 own personal computers.

a. Test at the 1% significance level whether the current percentage of all U.S. households who own personal computers is different from 32%.

b. Construct a 99% confidence interval for the true current proportion of all U.S. households who own personal computer.

6. During the past few years people have become health conscious, especially in regard to the consumption of red meat. In 1993, the average consumption of red meat per person was 111.9 pounds in the United States. A sample of 16 persons showed that they consumed, on average, 100.5 pound of red meat in 1996 with a standard deviation of 26.5 pounds.

b. Set up a 95% confidence interval for the true mean consumption of the red meat for 1996.