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1. (15 Marks) The following are body mass index (BMI) scores measured in 11 patients who are free of diabetes and participating in a study of risk factors for obesity. Body mass index is measured as the ratio of weight in kilograms to height in meters squared. 25, 27, 31, 28, 28, 41, 24, 32, 46, 35 and 40 Find the following: a) Mean (1 point) b) Median (1 point) c) Range (1 point) d) Lower Quartile (Q1) (1 point) e) Upper Quartile (Q3) (1 point) f) Interquartile Range (IQR) (1 point) g) Lower Fence (1 point) h) Upper Fence (1 point) i) Construct a box-plot. Are there potential outliers in the data? Justify your answer. (2 points) j) Compute the sample standard deviation and sample variance. (3 points) k) Which statistics would you use to identify the center and the spread of this distri- bution? Explain. (2 points) 2. (8 Marks - 2 points each) Suppose you toss a fair coin four times. Let the random variable X be the number of tails (T) obtained. Also, let E and V denote the mean and variance respectively. a) Compute E(X) and V(X). b) Compute E(5X - 6) and V(6 + 5X). 3. (9 Marks) The length of time required by students to complete a one-hour exam is a random variable with a probability density function (pdf) given by f(x) = c.x³ + x², 0 < X < 1. a) Find the value C and write down the new pdf. (2 points) C b) Find the probability that a randomly selected student will finish in less than half an hour. (3 points) c) Find the mean and standard deviation of the distribution. (4 points) mean = SD = 4. (8 Marks) Suppose X is a random variable best described by a uniformly distribution or probability that ranges from 2 to 11. a) Write down the probability density function f(x). (1.5 points) b) Compute the following: i) mean (1.5 points) ii) standard deviation (1.5 points) iii) P(X < 3.858) (1.5 points) iv) P(u - (2 points) 5. (5 Marks) At a certain gas station, 60% of the customers use Regular gas (A1), 20% use Plus gas (A2), and 20% use Premium (A3). Of those customers using regular gas, only 45% fill their tanks (event B). Of those customer using plus, 50% fill their tanks, whereas of those using premium, 55% fill their tanks. If the next customer fills the tank, what is the probability that Plus gas is requested? 6. (5 Marks) A group of four undergraduate and five graduate students are available to fill certain student government posts. If four students are to be randomly selected from this group, a) Find the probability that exactly two undergraduates will be among the four chosen. (2 points) b) What is the probability that at least two graduate students be chosen? (3 points)

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