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STAT 2000
Fall 1999
FINAL EXAM
NAME: __________________________
The total number of points for this exam is 400. The results of this exam will account for 40% of your total course grade. The exam consists of 7 problems. The maximum number of points for each problem and for each partial question is given in parentheses.
You have one hour and 50 minutes to complete this exam. Read the questions carefully. Start with the question which is easiest for you and then move on to more difficult problems.
Write in the spaces provided below each question. If necessary, write also on the back. Write neatly.
Show as much work as possible in order to get the full number of points. Write down each step that you are taking.
Some of the questions are multi-part questions, in which an answer to part a) is needed to answer the following parts. If you cannot answer part a) and you feel that you cannot proceed without knowing this answer, make up a (plausible) answer and proceed to part b) etc.
Problem I (70 points)
A friend who hears that you are taking a statistics course asks you for help with a chemistry lab report. She has made four independent measurements of the specific gravity of a compound. The results are:
3.82, 3.93, 3.67, 3.78.
The lab manual says that repeated measurements will vary according to a normal distribution with standard deviation σ = 0.15. The mean μ of the measurements is the true specific gravity.
(a) The lab manual asks for a 95% confidence interval for the true specific gravity. Your friend doesn't know how to do this. Do it for her! (20 points)
(b) Now explain to your friend in a simple language what "95% confidence interval" means. (10 points)
(c) What critical value of z *^ (from the last row of Table D) would you use if you wanted a 80% confidence interval instead of a 95% confidence interval? Would the 80% confidence interval be wider or narrower than your confidence interval from (a)? [Do not actually compute the 80% confidence interval]. (10 points)
Problem II (50 points)
The weights of newborn children in the United States vary according to the normal distribution with mean 7.5 pounds and standard deviation 1.25 pounds. The government classifies a newborn as having low birth weight if the weight is less than 5.5 pounds.
(a) What is the probability that a baby chosen at random weighs less than 5.5 pounds at birth? (25 points)
(b) You choose 3 babies at random. What is the probability that their average birthweight is less than 5.5 pounds? (25 points)
Problem III (70 points)
A scatterplot below illustrates the grade-point averages (GPAs) and verbal scholastic achievement test (SAT) scores for a sample of 100 students at a university in the northeastern United States.
(a) Describe the relationships between the GPAs and the SAT scores in a few sentences. Are there any outliers? If so, circle them. Guess an approximate correlation coefficient. (20 points)
Problem IV (60 points)
According to the U.S. Census data, 42% of the 1,722,900 residents of Utah in 1990 were under 21 years of age.
This year (1999), you want to conduct a survey on population issues in Utah. You choose a random sample of 1,000 Utah residents. You find out that 455 people in your sample are under 21 years of age.
(a) Does your sample provide evidence that the proportion of Utah residents younger than 21 years has increased since 1990 when the Census data were obtained? State the null hypothesis H 0 that the proportion of people under 21 years in 1999 has remained the same as the proportion obtained in 1990 and the alternative Ha that the proportion of people under 21 years has increased. Calculate the z statistic for testing H 0 versus Ha and find the P -value. Report your conclusions. (30 points)
(b) Give a 95% confidence interval for the proportion p of Utah residents who are under 21 years of age. (20 points)
(c) The test and confidence interval calculations that you performed are based on the normal approximation to the binomial distribution. Explain carefully why you could use the normal approximation in this particular problem. (10 points)
Problem V (40 points)
A life insurance company sells a term insurance policy to a 21-year old male that pays $100,000 if the insured dies within the next 5 years. The company collects a premium of $250 each year as payment for the insurance. The amount X that the company earns on this policy is $250 per year, less the $100,000 that it must pay if the insured dies.
The table below shows the distribution of X :
Age at death (^21 22 23 24 25) ≥ 26
Payout (^) −$99,750 −$99,500 −$99,250 −$99,000 −$98,750 $1,
Probability 0.00183 0.00186 0.00189 0.00191 0.00193?
(a) Fill in the missing probability in the table and calculate the mean earnings μ X. ( points)
(b) Under the terms given above, it would be quite risky for you as a life insurance agent to insure the life of only one 21-year-old person. There is a high probability that this person would live and you would gain $1,250 in premiums. But if this person were to die you would lose almost $100,000. Explain carefully why selling insurance is not risky for an insurance company that insures many thousands of 21-year-old men. ( points)
Problem VII (50 points)
Choose an American worker at random and classify his or her occupation into one of the following classes. These classes are used in government employment data.
A Managerial and professional B Technical, sales, administrative support C Service occupations D Precision production, craft, and repair E Operators, fabricators, and laborers F Farming, forestry, and fishing
The table below gives the probabilities that a randomly chosen worker falls into each of 12 gender-by-occupation classes:
Class A B C D E F
Male 0.14 0.11 0.06 0.11 0.12 0.
Female 0.09 0.20 0.08 0.01 0.04 0.
(a) Verify that this is a legitimate statement of probabilities to these outcomes. ( points)
(b) What is the probability that the worker is female? (10 points)
(c) What is the probability that the worker is not engaged in farming, forestry, or fishing? (10 points)
(d) Classes D and F include most mechanical and factory jobs. What is the probability that the worker holds a job in one of these classes? (10 points)
(e) What is the probability that the worker does not hold a job in classes D or E? ( points)
