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Statistics 350 Midterm 1 Time: 1 hour Name (please print): ___________________________ Show all your work and calculations. Keep 2 decimal places and circle your final answer. Partial credit will be given for work that is partially correct. Points will be deducted for false statements shown in the solution to a problem, even if the final answer is correct. This exam is closed-book. You may not consult any notes or books during this exam. Only a non-graphical calculator, a formula sheet and relevant tables are allowed. Please sign below to indicate your agreement with the following honour code. Honour code: I promise not to cheat on this exam. I will neither give nor receive any unauthorized assistance. I promise not to share information about this exam with anyone who may be taking it at a different time. I have not been told anything about the exam by someone who has already taken it. Signature: . Question Possible 1 8 2 16 3 10 4 16 Total 50 Actual Date: . 1. (20pts) Multiple choice and short calculation problems Consider a sample x1 , ..., xn with mean x and standard deviation s and let zi xi x , verify s the following statement in 1) , 2) and 3): 1) The mean of zi is 0 A) True B) False 2) The standard deviation of the zi is 1 A) True B) False 3) The zi ’s histogram is bell-shaped. A) True B) False Case study 4) If a random variable X denotes the number of trees in 2-acre plot within a certain forest, if an average density is 80 trees per acre, then which of the following statement is true A) X ~ Binomial (80, 0.5) B) X ~ Binomial (160, 0.5) C) X ~ Poisson(80) D) X ~ Poisson(160) 1 out of 5 2. Fill in the blanks, 2 points each, no partial credit. 1) If two random variables, X ~ Normal ( , ) , Y ~ Normal ( , ) ， X and Y are independent, find the answers to the following questions. E(X+Y)=_________________, E(X-Y)=______________________ Var(X+Y)=________________, Var(X-Y)=_______________________ Var(X+X)=__________________, Var(X-X)=____________________________ 2) Consider a system of components connected as shown in the following figure. Suppose all components work independently of one another and Pr(a given component works)=0.9, the probability that the entire system works correctly =_______________ 3) Suppose a random variable X follows a distribution of Binomial(4, 0.9), the median is ______________(<, > or =) the mean of the distribution . 2 out of 5 3.(10pt) The distribution of bone mineral density in women between the ages of 45 and 64 is normal with mean 0.9 g/ cm 2 and a standard deviation of 0.15 g/ cm 2 . a). (2pt) What percent of the population has bone mineral density more than 1.2 g/ cm 2 ? b) (2pt) What is the probability that the bone mineral density of a randomly chosen women is between 0.9 and 1.2 g / cm2 c) (3pt) Suppose that the bone mineral density of a randomly chosen woman is more than 0.9 g / cm2 , what is the probability that her bone mineral density is less than 1.2 g / cm2 ? d) (3pt) In this population, what is the bone mineral density such that only 3% of the population has a lower level? 3 out of 5 4. (16pt) A random variable X has a probability density function as follows: kx 2 (1 x) 0 x 1 f ( x) { 0 otherwise a) (2pt) Find the value of k . b) (2pt) Find the probability that X is greater than 0.5. c) (2pt) Find the mean of X . d) (2pt) Find the standard deviation of X . 4 out of 5 e) (4pt) Consider a randomly selected sample of size 20, find the probability that at least one observation is greater than 0.5. f) (4pt) Consider a randomly selected sample of size 20, find the proability that the sample mean is inclusively within 0.01 of 0.6 5 out of 5