Midterm1 Sample

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Statistics: Measurement in Economics Econ 310, Fall 2023 Sample First Midterm Answer as succinctly as possible while still fully answering the question. Note that a “yes” or “no” is never a complete answer – you are always expected to explain your reasoning. Solutions will be discussed at our in-class review session on Tuesday, Oct 10 and will be posted to Canvas shortly thereafter. Feel free to reference Appendix B of the textbook. (On an actual exam, any necessary tables would be provided.) Section A It is not necessary to explain your reasoning or show your work for this section. Each numbered question in this section is worth 3 points. 1) A researcher at Florida International University (FIU) wants to estimate the average number of credits earned by students last semester at FIU. In her random sample of 750 students, the average number of credits is 13.75. The population of interest to the researcher is: a. all FIU students b. all college students c. all FIU students enrolled last semester d. the 750 FIU students selected at random 2) Which of the following statements is true for the following observations? 9, 8, 7, 9, 6, 11, and 13 a. The mean, median and mode are all equal b. Only the mean and median are equal c. Only the mean and mode are equal d. Only the median and mode are equal 3) You have collected some height data. Assume height has a bell-shaped distribution with a mean of 66 inches and a standard deviation of 3 inches. Approximately what percentage of the individuals in your data will have heights between 60 and 72 inches? a. 68% b. 75% c. 95% d. 99% 4) If a researcher counts and records the number of students wearing backpacks on campus in a given day, what method of data collection is she using? a. Administrative data b. A survey c. Direct observation d. None of these choices

5) If A and B are mutually exclusive events with P(A) = 0.75, then P(B): a. can be any value between 0 and 1 b. can be any value between 0 and 0.75 c. can be any value between 0 and 0.25 d. can be any value between 0.25 and 1 e. equals 0.25 6) If A and B are exhaustive events with P(A) = 0.75, then P(B): a. can be any value between 0 and 1 b. can be any value between 0 and 0.75 c. can be any value between 0 and 0.25 d. can be any value between 0.25 and 1 e. equals 0.25 7) Which of the following best describes the concept of marginal probability? a. It is a measure of the likelihood that a particular event will occur, regardless of whether another event occurs b. It is a measure of the likelihood that a particular event will occur, if another event has already occurred c. It is a measure of the likelihood of the simultaneous occurrence of two or more events d. None of these choices 8) Which of the following is false? a. E(3X) = 3E(X) b. V(2) = 2 c. E(X + 1) = E(X) + 1 d. Cov(2X,Y) = 2Cov(X,Y) e. All of these choices are true 9) The covariance of two variables X and Y: a. must be between − 1 and +1 b. must be non-negative c. can be any real number d. None of these choices 10) In a Poisson distribution, the: a. mean equals the standard deviation b. median equals the standard deviation c. mean equals the variance d. None of these choices

11) Suppose f(x) = 0.25. What range of possible values can X take on and still have the density function be legitimate? a. [0, 4] b. [4, 8] c. [ − 2, +2] d. All of these choices are true 12) The distribution in the previous question is known as a: a. normal distribution b. uniform distribution c. Poisson distribution d. binomial distribution Section B Each lettered question in this section is worth 5 points Assume 6% of the world’s population has a particular genetic condition. A blood test for the condition is available, but it is not 100% accurate. For those who do not have the genetic condition, the blood test will return a positive result 3% of the time. For those who do have the genetic condition, the test returns a negative result 8% of the time. Consider the random experiment that an individual is randomly selected and tested for the genetic condition. Let C be the event that the individual has the genetic condition, NC be the event that they do not. Let PT be the event that the blood test is positive and NT be the event that it is not. a) Draw a probability tree for this random experiment b) What is the probability the individual will test negative for the genetic condition? c) What is the prior probability that the individual has the condition? d) Suppose the test result is negative. What is the posterior probability the individual has the condition?

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Section C Each lettered question in this section is worth 4 points A student is taking a quiz with 12 questions. Each question is equally difficult and the student has studied a lot, so the probability of getting each individual question correct is 0.9. Consider the number of questions this student gets correct on the quiz. a) Does this random variable (the number of correct answers) satisfy the assumptions of the binomial distribution? Explain your reasoning. b) For the remainder of this section, suppose the number of correct answers does satisfy the assumptions for a binomial distribution. What is the probability the student gets exactly 7 questions correct? c) Now suppose the quiz is shortened to 10 questions and that it’s necessary to get 5 or more questions correct to pass the quiz. What is the probability the student passes? Section D Each lettered question in this section is worth 5 points Let X be a random variable with probability mass function: f(x)=c/x if x=1,2,3 a) What value must c take for this to be a proper probability mass function b) Carefully graph f(x). What is f(x) equal to when x is not equal to 1, 2, or 3? Carefully graph the cumulative distribution function for X c) Calculate the mean and variance of X d) What is P(1<X ≤ 3) Section E Each lettered question in this section is worth 3 points Suppose you have a random variable W with μ w =1 and σ w =1, a second random variable X with μ x =2 and σ x =2, and a third random variable Z with μ z =3 and σ z =3. Note that σ wx =0 and σ wz =-1 a) What is E(10W+10X+10)? b) What is V(10W+Z+10)? c) Are W and Z independent? Are W and X independent?

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