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ST361 HW5: Practice Problems of M1 Notice: (1) No need to turn in (2) You may use one sheet of notes (8.5 by 11 inches) and a calculator. You may not share a calculator, pencil, paper or anything else during the exam. Part I: Multiple Choice Questions (3 points each) ___________ 1. If P(B|A) = P(B), then events A and B are said to be (a) complementary (b) disjoint (c) independent (d) conditional ___________ 2. Which of the following statement is not true about a normal curve? (a) Every normal curve has its under-curve area equal to 1 (b) Every normal curve has its standard deviation (SD) equal to 1 (c) Every normal curve has its mean equal to its median (d) Every normal curve is symmetric about its mean (e) Every normal curve is bell-shaped ___________ 3. Variable Z follows the standard normal distribution. If the area to the right of z* is 0.0985, then the value of z* must be (a) 0.84 (b) -1.36 (c) 1.36 (d) 1.29 (e) -1.29 . ___________ 4. Variable Z follows the standard normal distribution. What is P 1 Z 1 ? (a) 0.6826 (b) 0.8413 (c) 0.3174 (d) 0.1587 (e) 0.1357 1 Questions 5 to 7. The histogram below describes of the lifetime (in hours) of a certain type of lamp: ___________ 5. Which of the following best describes the shape of the histogram? (a) Symmetric (b) Positively skewed (c) Negatively skewed (d) None of the above ___________ 6. For the lifetime data described by the above histogram: (a) The median will be bigger than the mean (b) The median will be smaller than the mean (c) The median and the mean will be about the same (d) The median and the inter-quartile range will be about the same (e) The median will be smaller than the inter-quartile range ___________ 7. Which of the following summary measures should be used to describe the lifetime data? (a) Mean and SD (b) Mean and IQR (c) Median and inter-quartile range (IQR) (d) Median and standard deviation (SD) 2 ___________ 8. Data were collected on the body temperatures of the 76 sick-visit children at a pediatrics clinic during a month. The temperature was measured in Fahrenheit ( F ). If the temperature has been measured in Centigrade ( C ), which of the following summary measures (in C ) would be changed comparing to their values in F ? (Note that C F 32 5 .) 9 (a) (b) (c) (d) (e) The mean temperature The median of the temperature The IQR of the temperature (a) and (b) (a),(b) and (c) ___________ 9. The probability mass function of a discrete random variable x is defined as p(x) = cx for x = 1,2,3,4, then the value of c is (a) 0.01 (b) 0.05 (c) 0.10 (d) 0.20 (e) 0.40 ___________ 10. A continuous random variable X has the following density plot: What is the probability of X between 0 and 3? (a) 0.0 (b) 0.4 (c) 0.5 (d) 0.7 (e) 1.0 3 ___________ 11. Approximately 14 percent of the population of Arizona is 65 years or older. A random sample of five persons from this population is taken. The probability that less than 2 of the 5 are 65 years or older is (a) 0.8533 (b) 0.1467 (c) 0.4704 (d) 0.3829 (e) None of the above. ___________ 12. A barbershop requires appointments for perms and hair cuts, and about 8 % of those appointments tend to be canceled. Next week's appointment calendar has 60 appointments. Let X be the number of missed appointments out of the 60. What is the expected number of cancellations occur during the next week? (a) 4.8 (b) 6 (c) 8 (d) 10 (e) 60 ___________ 13. In the following density plot, the solid line A is the density curve for a standard normal distribution. Which of the following curve is the density curve for N(-2.5, 1)? (a) Curve A (b) Curve B (c) Curve C (d) Curve D 4 ___________ 14. If P(A) = 0.3, P(B) = 0.4, and events A and B are independent. Then P(A or B) is (a) 0.12 (b) 0.58 (c) 0.60 (d) 0.70 (e) 0.80 Questions 14 to 15. John works in a big city and owns two cars, one small and one large. 70% of the time he drives the small car to work, and 30% of the time he drives the large car. If he drives the small car, he usually has little trouble parking, and hence is on time to work with probability 0.8. If he drives the large car, he is on time to work with probability 0.6. ___________ 15. What is the probability that John is on time at work and drives his large car? (a) 0.18 (b) 0.21 (c) 0.24 (d) 0.42 (e) 0.56 ___________ 16. Assume the probability that John is on time to work is 0.74. Given that he was on time in a particular morning, what is the probability that he drove the large car? (a) 0.24 (b) 0.28 (c) 0.32 (d) 0.57 (e) 0.76 5 Part II: Fill in The Blank Assume a data set is presented the stem-and-leaf plot below (sample size =18): 1|1 2 | 468 3 | 022345689 Stem: 1 4 | 0578 Leaf: 0.1 5| 6|1 Based on this data set, Mary and Kate constructed the boxplots below (Note a = minimum). 6 1. In the plots above, “b” = ________________, and “d” = ________________ (Report their values; 3 points each) 2. The inter-quartile range (IQR) is _________________ (3 points) 3. Use the 1.5 IQR rule to check if any outliers exist. Report your criteria for outliers, and the values of outliers if any. (7 points) 4. Based on your conclusion in part 3, whose boxplot do you agree with? (3 points) _______________ Mary’s _______________ Kate’s _______________ None of them _______________ Both of them 7 Part III: Computational Problems Question 1. A system consists of 2 components in parallel. The components work or fail independently of one another, and each component works with probability 0.8. Let A1 denote the event that the first component works, and A2 denote the event that the second component works. Assume the system works as long as at least one component works. (a) Shade the region on the Venn diagram corresponding to the event of “both components work”. (3 points) S A1 A2 (b) Calculate the probability that both components work. (4 points) (c) Shad the region that corresponds to the event that “at least one component works” (3 points) S A1 A2 (d) Calculate the probability that at least one component works. (4 points) 8 Question 1 (continued) (e) Shad the region that corresponds to the event that “the system does not work” (3 points) S A1 A2 (f) Calculate the probability that the system does not work. (4 points) Partial answers of practice problems: CBDA CAC ECD AAB BAA Part II: 3.0;4.0;1; any numbers <1.5 or >5.5 are outliers; 1.1 and 6.1 are outliers; Mary’s Question 1 (b) 0.64; (d) 0.96; (f) 0.04 Question 2: 0.242; 0.6612; 101.8 9 Question 2. Grades of students are given according to the exam scores which range from 0 to 120. A student will get an “A” if the score is equal or above 90 points, “B” if the score is equal or above 70 points but less than 90 points, and “C” if the score is less than 70 points. The scores on the exam have a normal distribution with mean 83 and standard deviation 10 points. (a) Find the proportion of students will get “A”. (5 points) (b) Find the proportion of students will get “B” (5 points) (c) Assume now that another category “A+” was introduced, which includes the students whose score on the test was in the top 3 %. A student with grade “A+” implies the score is great than _______________ points on the test. (5 points) 10