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STAT 3000
Spring 2002
FINAL EXAM
NAME: ____________________________
The total number of points for this exam is 300. Your grade on this exam will account for about 21% of your total course grade.
The exam consists of 5 “regular” problems and one “extra credit” question. The maximum number of points for each problem and for each partial question is given in parentheses.
Write in the space provided below each question. If necessary, write also on the back. Write neatly.
Show your work in order to get the full number of points. Write down each step you are taking. Just the end result, even if it is entirely correct, is not enough.
You have about two hours (exactly: one hour and 50 minutes) to complete this exam. Read the questions carefully. Start with the question which seems easiest for you and then move on to more difficult problems.
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: Parameters and Statistics (50 points)
1. For each of the following questions, state whether each underlined number is a parameter or a statistic.
a) (10 pts) A shipment of 1000 fuses contains 3 defective fuses. A sample of 25 fuses from this shipment contains 0 defectives.
b) (5 pts) The speed of a random sample of 100 vehicles on I-15 was monitored. It was found that 63 vehicles exceeded the posted speed limit.
c) (10 pts) A telephone poll of registered voters one week before a statewide election showed that 48% would vote for the current governor, who was running for reelection. The final election returns showed that the incumbent won with 52% of the votes cast.
2. (15 pts) A random sample of 10 coffee cans is taken from a production line and their contents are weighted. The weights (in oz.) are as follows:
26.3, 25.9, 26.9, 26.7, 25.8, 25.3, 26.4, 25.7, 26.3, 26.
Calculate the point estimate of the population mean and the standard error of this estimate.
3. (10 pts) At the beginning of Bill Clinton’s presidential administration, he spoke on proposed economic reforms. A sample of people who heard the speech ( n = 611) were asked if they favored higher taxes on all forms of energy: 43% responded “Yes”. What is the point estimate of the population proportion of people who favored higher energy taxes? What is the standard error of this estimate?
Problem III: Comparing Two Population Means: Independent Samples (80 points)
A study was made to determine if the subject matter in a physics course is better understood when a lab constitutes a part of the course. Students ware randomly selected to participate in either a 3-credit course without labs or a 4-credit course with labs. In the section with labs 11 students made an average score of 85 points, with a standard deviation of 4.7, and in the section without labs 16 students made an average score of 79 points, with a standard deviation of 6.1.
a) (40 pts) Would you say that the laboratory course increased the average score by as much as 5 points? Clearly state H 0 and HA and carry out an appropriate statistical test assuming unequal population variances. Give the (approximate) p -value and state your conclusions. Explain your findings in your own words without using statistical terminology.
b) (20 pts) Construct a 95% one-sided confidence interval for the lower bound of the mean difference μA − μB of the average scores in courses with and without labs, again assuming unequal population variances. How does this confidence interval support your conclusions from part a)?
c) (20 pts) Would it be safe to use the pooled variance procedure for this problem? To find this out, carry out an F - test to check whether the variances in these two populations (consisting of scores of students taking a course with and without labs, respectively) could be assumed equal. Then just answer the question above – do not perform a pooled variance procedure (even if the answer to the question above is positive).
Problem V: Sampling Distributions (60 points)
Consider a sample X 1 , ….. Xn of normally distributed random variables with mean μ and variance σ^2 = 5. Suppose n = 25.
a) (15 pts) What is the probability that
μ − X ≤ 0_._ 5?
b) (15 pts) What is the value of c for which
P( S^2 ≤ c) = 0_._ 90?
c) (15 pts ) Let
S
X
T
− μ
.What is the value of c for which
P^ (^ T ≥ c )^ = 0. 10?
d) (15 pts) Let
S
X
T
− μ
, U = T^2. What is the distribution of U?
Problem VI: Extra Credit Question (40 points)
Adapted from “Sample Final” by Moto Machida, Fall 2000
On the first day of a new job, you were asked to accompany the executive to a biomedical research firm. This firm has developed a new penicillin manufacturing process, claimed to be better than the process your company is currently using. Thus, your company is considering buying a license for using this new method. This is a serious decision for your company: the license for using the new method is very expensive, but if the new method really works as advertised, your company would be able to produce more penicillin per manufacturing unit and the profit would be huge. The executive ordered his staff to investigate on how well this new method works compared to the current method. Here is the summary of the report produced by his staff:
To determine the effect of the new method on the yield of penicillin, the data were collected for five types of base blend (B1 to B5) to produce penicillin. “Method I” refers to our current process and “Method II” refers to the process newly developed by the biomedical firm. Penicillin yields are presented in the following table:
Blend Method I Method II B1 89 97 B2 84 92 B3 83 87 B4 87 89 B5 80 79
For statistical inference, the following hypotheses were tested: H 0 : μI ≥ μII vs HA: μI < μII where μI and μII are population means for Method I and Method II, respectively. An inference on two independent samples was considered and the details of the statistical analysis are given below:
t-Test: Two-Sample Assuming Unequal Variances
Method I Method II Mean 84.6 88. Variance 12.3 44. Observations 5 5 Hypothesized Mean Difference 0 df 6 t Stat -1. P(T<=t) one-tail 0. t Critical one-tail 1. P(T<=t) two-tail 0. t Critical two-tail 2.
