Math 201: Statistics November 30, 2006

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1 Math 201: Statistics November 30, 2006 Fall 2006 MidTerm #2 Closed book & notes; only an A4-size formula sheet and a calculator allowed; 90 mins. No questions accepted! Instructions: There are eleven pages (one cover, seven pages with questions and three pages with probability tables) in this exam. Please inspect the exam and make sure you have all 11 pages. You may only use your calculator and your formula sheet. Do all your work on these pages. If you use the back of a page, make sure to indicate that. You may not exchange any kind of material with another student. Remember: You must show all your work to get proper credit. Academic Honesty Code: Koç University Academic Honesty Code stipulates that copying from others or providing answers or information, written or oral, to others is cheating. By taking this exam, you are assuming full responsibility for observing the Academic Honesty Code. NAME and SURNAME: SIGNATURE: INSTRUCTOR : LECTURE TIME : Total:

2 TRUE/FALSE: Use a capital T for true and a capital F for false. (2 points each) 1. For a confidence level of 93%, the confidence interval using the t-distribution is larger than the interval using the z-distribution. 2. The population mean changes with the confidence interval. 3. Sample proportion is not equal to sample mean. 4. The standard error decreases as the sample size increases. 5. For a general normal distribution, each z value is the number of standard deviations from the mean. 6. The number of ways that 3 heads and 2 tails can be arranged is The area under the curve for a normal distribution is equal to one. 8. Two random variables, X and Y, are independent if X is not equal to Y. 9. If a 90 % confidence interval is between and 10.0, the sample mean is A population can be any size. 11. We use the t-distribution when the population standard deviation is not known. 12. The sample mean is an unbiased estimator of the population mean.

3 SHORT CALCULATIONS: Place your answer on the blank next to the problem number and show your work. 13. What should be the sample size from a population if one wants the 90 % desired margin of error to be 2? The standard deviation of the population is 10 and the mean of the population is 5. (4 points) 14. From a large class, a random sample of 2 grades were drawn: 40 and 60. Calculate a 95% confidence interval for the mean of the whole class. (4 points) 15. Find z o such that P(-z o <Z<z o )=0.9 (3 points) 16. Calculate the probability of having exactly one boy in a family of 3 children if each child has a 50% chance of being a boy. (3 points) 17. If X is normally distributed around a mean of 10 with a standard deviation of 2, find P(6<X<12). (3 points)

4 PROBLEMS: Show all your work to be eligible for partial credit. Round final answers to three decimal places. 18. The Agamama restaurant is planning to designate a smoking area in view of the fact that the chance of a randomly selected customer being a smoker is The seating capacity of the restaurant is 90. (a) What is the probability that 2 are smokers when there are only 7 customers in the restaurant? (3 points) (b) Suppose 33 seats are allocated to the smoking area. What is the (approximate) probability that all smokers can find a seat in this area in a full capacity evening? (5 points) (c) The profit of the restaurant from a customer is normally distributed with mean 10 YTL and standard deviation 2 YTL, independent of smoking habits. What is the probability that a randomly selected customer is a non-smoker and brings more than 12 YTL profit? (5 points)

5 19. The life insurance policy of Trust Insurance Co. stipulates that the policyholder will receive a payment on his/her 60 th birthday, and a payment every five years thereafter. Suppose lifetimes of this insurance company s policyholders are normally distributed with mean 63 years and standard deviation 4 years. (a) Let X = Lifetime of a policyholder from this insurance company. Find the 86 th percentile of X. (3 points) (b) What proportion of policyholders will receive at least one payment? (4 points) (c) What proportion of policyholders will receive exactly 2 payments? (4 points) (d) In a random sample of 9 policyholders, what is the probability that the average individual will receive at least one payment? (6 points)

6 20. A consultant reported the following data from small guesthouses. In the table, X = Number of persons served, Y = Average daily cost (YTL) per person, and the numbers in the cells are relative frequencies, which may be interpreted as probabilities. X = 4 X = 6 X = 8 Y = Y = Y = (a) What is the expected average daily cost per person? (4 points) (b) What is the probability that average daily cost per person is more than 15 YTL given that the number of persons served equals 4? (4 points) (c) Are X and Y independent? Show. (4 points)

7 21. The probability distribution of X = Number of cars sold in a month by a car dealer is given below: x p(x) (a) Find the mean and standard deviation of X. (4 points) (b) What is the probability that the total number of cars sold in two months is equal to 30? Assume the number of cars sold on different months are independent. (4 points) (c) What is the approximate probability that the car dealer will sell more than 720 cars in 48 months? (6 points)

8 22. Dr. Glowy wants to estimate how many patients she treats in a day. (a) She takes a random sample of 12 days and assumes that the number of patients she treats in a day has a normal distribution. She finds that the sample mean is 40 and the standard deviation is 8 for this sample. Construct a 90% confidence interval for the mean number of patients treated in a day by using these sample statistics. (5 points) (b) If she takes another random sample of 12 days, will the 90% confidence interval change? Why or why not? (3 points) (c) For a new random sample of 60 days, she found the sample mean and sample standard deviation same as in part (a). Could she calculate now the 90% confidence interval for this sample, if she is sure that the number of patients she treats in a day does not have a normal distribution? If yes, construct the 90% confidence interval. (5 points)

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