Section I Questions with parts

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1 Name: SFU ID: Stat 201 Midterm Solutions page 1 Section I Questions with parts Most parts in these questions are related. For a part that you can t do, write Can t do -- suppose answer is and fill in some arbitrary answer. You will not lose marks for using this arbitrary answer in the subsequent parts. Question 1 [14 points] A study is administered to assess the effectiveness of the new Benny Cake weight loss program. The 400 subjects 158 males and 242 females enrolled in the study are weighed before the start of the program. Then each goes on a diet of five meals of Benny Cakes on a daily basis. At the end of a six-month period, each subject is weighed again, and the weight loss BEFORE - AFTER is recorded. A negative weight loss means weight gain. a[1 ] The design of this study employs blocking. What is the blocking factor? the subject i.e. matched-pair b[1 ] Suggest another blocking factor and explain in no more than 20 words why you d wish to block on this factor. gender.5 pt: males and females may respond differently to the diet.5 pt c[2 ] Is this an experiment or observational study? Underline the correct answer and explain in no more than 20 words. experiment 1 pt: each subject was put to undergo treatment diet and put to be his/her own control the before measurement something to this effect --- note that there were no existing treatment and control groups to be observed --- gets you 1 pt

2 Name: SFU ID: Stat 201 Midterm Solutions page 2 d The data collected on the 400 subjects without the blocking factor in b are summarized as follows: weight lb weight lb weight loss lb before after BEFORE - AFTER mean SD i[5 ] Find a 95% C.I. for the mean weight loss. Do you need assumptions for this C.I. to be valid? Why? Use no more than 20 words. n = 400 huge, so s d σ d and use z-score: d ± 2 s d n 1 = 0.5 ± = 0.5 ± = 0.5 ± = 0.3, No assumptions are necessary, as n = 400 is large enough to apply CLT and have s d σ d 1 pt line 1 alone is 1 pt; combination of lines 1 to 5 with correct substitution of values is 4 pts; each correct value for d, s d, n, and multiplier which can also be 1.96 or 1.97 is 1 pt each ii[2 ] Based on the answers in c and di, what can you say about the effectiveness of the Benny Cake program in reducing weight? Use no more than 20 words. If answer in c is experiment : We are 95% confident that the program caused a mean weight loss of between 0.3 and 0.7 pouunds 1 pt, i.e. effective 1 pt. Note: we conclude causation because this is a matched-pair experiment and not observational. But marks for this question are given according to your answer in c. If answer in c is observational : We are 95% confident that the mean weight loss is between 0.3 and 0.7 pounds 1 pt, but we can t say the program is effective because the study is observational 1 pt. iii[3 ] If you re told the population mean BEFORE weights is lb and the SD is 20 lb, what is the probability that a random sample of 400 subjects walk into the Benny Cake program averaging less than 200 lbs? Let X i be weight of i-th person walking into the program. n large; use CLT: Nµ, σ x / n 6 X µ P X < 200 = P σ x / n < 200 µ σ x / 7 n CLT P Z < 20/ = P Z < 1.5 = P Z < 1.5 = X approx

3 Name: SFU ID: Stat 201 Midterm Solutions page 3 combination of lines 6 and 7 alone is 1 pt; line 8 alone or correct substitution of all values is 1 pt; correct prob. value from normal table alone is 1 pt; combination of lines 6 to 9 is 3 pts Question 2 [15 points] There are two very large schools, A and B. It is known that the SD of the children s IQ s are about the same for both schools. The goal is to determine which School produces more intelligent students. Now I obtain a random sample of 10 children from each school and observe the following: a[4 ] Identify the following: mean SD Sample A Sample B i Populations: Children of Schools A & B ii Samples: the 10 children selected from each of Schools A & B iii Parameter of interest Hint: just ONE: difference between Schools mean IQ s OR mean IQ for School A minus mean IQ for School B iv Statistic corresponding to iii: OR if answer for parameter is mean B minus mean A b[5 ] Use the pooled method to compute a 95% confidence interval for the difference between the mean IQ s for the two schools. State all assumptions necessary for the C.I. to be valid. NOTE: If you don t have a calculator, write down what you think is closest to the actual value. E.g = 4 Assume normal distribution for both schools IQ scores 1 pt with common SD 1 pt. n x 1 s 2 x + n y 1 s 2 y 1 pt s p = 10 n x + n y = = { 81 = 9 w/out calculator = 9.06 w/ calculator 11 1 pt t = since df=9+9= pt x y ± t 1 s p n x n y = ± /10 + 1/ ,

4 Name: SFU ID: Stat 201 Midterm Solutions page 4 c[2 ] Based on your answer in b, what can you say about either school producing more intelligent students, and why? Answer in no more than 30 words. We are confident School B produces more intelligent students 1 pt since we are 95% confident the difference between the mean IQ s for Schools A and B is between -19 and -1 less than 0 1 pt. d[1 ] Why should we use the pooled method in b? Use no more than 20 words. We are told the population SD s are very close. e[3 ] Circle all the true statements from below: i The central limit theorem applies because both schools are very large. ii It is known that the mean IQ for School A students is 110, and 115 for School B students; thus, switching a child from School A to School B will likely increase his/her IQ. iii If we repeatedly perform the same sampling procedure as stated, then about 95% of these operations will produce a 95% C.I. that includes the difference in mean IQ between the two schools. Section II Multiple Choice questions [8 points] For each of the following questions, circle the correct option only ONE. A correct answer is worth 2 points, and each wrong answer is worth 0 points. You need NOT show work. 3. The weather reporter says: The probability of showers on Saturday is 50%, and on Sunday is also 50%. It looks like 100% chance of rain sometime on the weekend. In the context of the weather report, which statement below is true only ONE? a You can add the probabilities because the events are independent. b You can add the probabilities because the events are mutually exclusive. c You can t add the probabilities because the events are not independent. d You can t add the probabilities because the events are not mutually exclusive. e none of the above

5 Name: SFU ID: Stat 201 Midterm Solutions page 5 4. Lie Detector. There is a probability of 0.3 that a suspect lies while undergoing the liedetector test. The test correctly detects a liar with probability 0.8, and correctly identifies an honest convict with a probability of 0.6. Given that the test result indicates a liar, what is the probability that the suspect actually lied? a 0.24 b c d e none of the above 5. Each Stat201 student rolls a six-face die. If s/he gets an ace 1 dot, then s/he will take a bonus exam. Thus, P take bonus exam = 1/6. There are 182 students in the class. The approximate probability that or more of them i.e. at least 25% will take the bonus exam is: 182 a b c P d P Z > Z > /6 1/6 5/ / e none of the above 6. Suppose a random sample of 100 safety pins from the manufacturer s warehouse showed a 12% of defective pins. The goal is to compute a 95% C.I. to estimate the true defective rate proportion of defective pins in the warehouse based on the estimate p=0.12. Which of the following statement is correct? a The t distribution should be used to find a 95% C.I. for p because the variance of the estimator has to be estimated from the data. b We can use the t distribution to find the C.I. because we can safely assume a normal population. c The Central Limit Theorem CLT ensures that the population is normally distributed so we can compute the C.I. based on the normal curve. d The C.I. computed based on p=0.12 has a 0.95 probability of caputuring the true p. e none of the above Congratulations! This is the end of the exam.

STAT 201 Assignment 6

STAT 201 Assignment 6 Partial Solutions 12.1 Research question: Do parents in the school district support the new education program? Parameter: p = proportion of all parents in the school district who