Exam-II+%283321%29_version1

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3321

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Feb 20, 2024

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Exam II BSTAT 3321 Last Name ........................................................... First Name .......................................................... ID# ...................................................................... Version 1 1 Please show the supporting work in the given space after each question (where applicable)! Answers without supporting work will not be given credit. You have 75 minutes to complete this exam. Total points is 100 .

1) [8] The weight of competition pumpkins at the Circleville Pumpkin Show in Circleville, Ohio, can be represented by a normal distribution with a mean of 703 pounds and a standard deviation of 347 pounds. a. Find the probability that a randomly selected pumpkin weighs at least 1,622 pounds. b. Find the probability that a randomly selected pumpkin weighs between 465.1 and 1,622 pounds. 2) [8] A telemarketer knows that, on average, he is able to make three sales in a 30-minute period. Suppose the number of sales he can make in a given time period is Poisson distributed. a. What is the probability that he makes exactly four sales in a 30-minute period? b. What is the probability that he makes at least two sales in a 30-minute period? Version 1 2

3) [9] An investor has a $120,000 portfolio of which $50,000 has been invested in Stock A and the remainder in Stock B. Other characteristics of the portfolio are as follows: Stock A Stock B E ( RA ) = μA = 11.2% E ( RB ) = μB = 8.6% σA = 15.93% σB = 9.65% Cov ( R A ,R B ) = σ AB = 38.20% a. Calculate the correlation coefficient. b. Calculate the expected return of the portfolio. c. Calculate the standard deviation of the portfolio. 4) [4] Stratified random sampling is preferred when the objective is to ______. 5) [7] Jennifer is waiting for a taxicab. The average wait time for a taxi is six minutes. Suppose the wait time is exponentially distributed. What is the probability that a taxi arrives in three minutes or less? Version 1 3

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6) [4] Which of the following can be represented by a discrete random variable? A) The circumference of a randomly generated circle B) The time of a flight between Chicago and New York C) The number of defective light bulbs in a sample of five D) The average distance achieved in a series of long jumps 7) [4] The t df distribution has broader tails than the z distribution. ⊚ true ⊚ false 8) [4] According the empirical rule for normally distributed variables, 75% of the values fall within one standard deviation of the mean. ⊚ true ⊚ false 9) [4] Nonresponse bias occurs when ______. A) the population has been divided into strata B) portions of the population are excluded from the sample C) cluster sampling is used instead of stratified random sampling D) those responding to a survey or poll differ systematically from the nonrespondents 10) [7] A sample of the weights of 35 babies is taken from the local hospital maternity ward. The point estimate for the mean weight of the babies is 110.2 ounces with a sample standard deviation of 23.4 ounces. Construct a 90% confidence interval for the population mean baby weight at this hospital. Interpret this interval. Version 1 4

11) [4] What is the purpose of calculating a confidence interval? A) To provide a range of values that has a certain large probability of containing the sample statistic of interest. B) To provide a range of values that, with a certain measure of confidence, contains the sample statistic of interest. C) To provide a range of values that, with a certain measure of confidence, contains the population parameter of interest. D) To provide a range of values that has a certain large probability of containing the population parameter of interest. 12) [4] A risk-averse consumer ignores risk and makes his or her decisions solely on the basis of expected value. ⊚ true ⊚ false 13) [8] The snowfall (in inches) during the month of January in a particular geographic region is normally distributed with a standard deviation of 16.75. In the last 12 years, the sample average of snowfall is computed as 122.50. a. Construct a 90% confidence interval of the average snowfall in this region. b. Construct a 95% confidence interval of the average snowfall in this region. 14) [4] For a given confidence level and population standard deviation, which of the following is true in the interval estimation of the population mean? A) If the sample size is bigger, the interval is narrower. B) If the sample size is smaller, the interval is narrower. C) If the population size is bigger, the interval is narrower. D) If the population size is smaller, the interval is narrower. 15) [4] The normal distribution is ______ in the sense that the tails get closer and closer to the horizontal axis but never touch it. Version 1 5

16) [8] Suppose your firm is buying five new computers. The manufacturer offers a warranty to replace any computer that breaks down within three years. Suppose there is a 25% chance that any given computer breaks down within three years. a. What is the probability that exactly one of the computers breaks down within five years? b. What is the probability that at least one of the computers breaks down within five years? 17) [9] The office of career services at a major university knows that 74% of its graduates find full-time positions in the field of their choosing within six months of graduation. Suppose the office of career services surveys 25 alumni six months after graduation. a. What is the probability that at least 80% of the alumni have a job in the field of their choosing? b. What is the probability that between 60% and 76% of the alumni have a job in the field of their choosing? c. What is the probability that fewer than 60% of the alumni have a job in the field of their choosing? Version 1 6

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Answer Key Test name: Exam-II (3321) 1) Short Answer a. 0.0040 b. 0.7495 The normal transformation implies that any value x of X has a corresponding value z of Z given by Use z table that provides cumulative probabilities P ( Z ≤ z ) for positive and negative values of z . a. Compute P ( X ≥ 1,622). Note that P ( Z > z ) = 1 − P ( Z < z ). b. Compute P (4,671.5 ≤ X ≤ 1,622). Note that P ( z 1 ≤ Z ≤ z 2 ) = P ( Z ≤ z 2 ) − P ( Z ≤ z 1 ). 2) Short Answer d. 0.1680 e. 0.8010 These results can also be found using Excel’s POISSON.DIST function. (See the text for details.) For a Poisson random variable X , the probability of x successes over a given interval of time or space is calculated as the mean number of successes or expected value; e ≈ 2.718. f. Calculate P(X = 4). g. Calculate P(X ≥ 2) = 1 − P(X = 0) − P(X = 1). Pay attention to the value of the time interval. The Excel's function POISSON.DIST can be used. 3) Short Answer h. 0.2485 i. 9.6833% j. 9.71% Version 1 7

The correlation coefficient between the returns R A and R B is calculated as An expected return of the portfolio E ( R p ) is calculated as E ( R p ) = W A E ( R A ) + W B E ( R B ); W A and W B are portfolio weights. The portfolio variance is calculated as The standard deviation of the portfolio is E ( R P ) = 11.2 × 0.42 + 8.6 × 0.58 = 9.6833 Var ( R P ) = 15.93 2 × 0.416667 2 + 8.65 2 × 0.583333 2 + 2 × 0.416667 × 0.583333 × 38.20 = 94.3134 SD ( R P ) = = 9.71 4) increase precision When the objective of sampling is to increase precision then the stratified random sampling is preferred. 5) Short Answer 0.3935 The cumulative distribution function for the exponential probability distribution is defined as P ( X ≤ x ) = 1− e −λx ; Compute P ( X ≤ 3). 6) C A discrete random variable assumes a countable number of possible values, whereas a continuous random variable is characterized by uncountable values. 7) TRUE The t df distribution has slightly broader tails than the z distribution. 8) FALSE Version 1 8

For normally distributed random variables the empirical rule states that 68.26% of the values fall within one standard deviation of the mean. 9) D Nonresponse bias refers to a systematic difference in preferences between respondents and nonrespondents to a survey or a pool. 10) Short Answer [103.5119, 116.8881] Ninety percent of similarly constructed intervals of size 35 would contain the population mean weight of babies in this hospital. Because the population standard deviation is unknown use t df distribution. The confidence interval of the population mean is computed as Use t table. 11) C The concept of confidence intervals is used to estimate the unknown population parameters. For a given calculated confidence interval, the probability of containing this parameter is actually either 1 or 0 because this interval may contain the parameter or not. 12) FALSE A risk-averse consumer incorporates risk in his or her decision to accept a risky prospect. 13) Short Answer k. [114.5466, 130.4534] l. [113.0230, 131.9770] <p>The confidence interval for the population mean is computed as Use z table. For a given sample size n and population standard deviation σ, the margin of error is greater, the greater the confidence level 100(1 − α )%. 14) A For a given confidence level and population standard deviation, the bigger the sample size n , the narrower the confidence interval. Population size is not part of the formula to create a confidence interval. 15) asymptotic Version 1 9

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The normal distribution is asymptotic. 16) Short Answer m. 0.3955 n. 0.7630 For a binomial random variable X , the probability of x successes in n Bernoulli trials is calculated as </p> o. Calculate P(X = 1). p. Calculate P(X ≥ 1) = 1 − P(X = 0). q. The expected value of the binomial probability distribution is computed as E(X) = μ = np. 17) Short Answer a. 0.2470 b. 0.5349 c. 0.0553 <p>If is normal, we can transform it into a standard normal random variable as , and any value of on has a corresponding value z on Z given by . a. Compute P ( ≥ 0.8). Note that P ( Z ≥ z ) = 1 − P ( Z < z ). Use z table. Compute P ( < 8). Use z table. The appropriate Excel function is =1-NORM.DIST(0.8,0.74,SQRT(0.74*(1-0.74)/25),TRUE) = 0.2470 b. Compute P (0.6 ≤ ≤ 0.76). Note that P ( z 1 ≤ Z ≤ z 2 ) = P ( Z ≤ z 2 ) − P ( Z ≤ z 1 ). Use z table. The appropriate Excel function is =NORM.DIST(0.76,0.74,SQRT(0.74*(1-0.74)/25),TRUE)- NORM.DIST(0.6,0.74,SQRT(0.74*(1-0.74)/25),TRUE) = 0.5349 c. Compute P ( < 0.6). Use z table. The appropriate Excel function is =NORM.DIST(0.6,0.74,SQRT(0.74*(1-0.74)/25),TRUE) = 0.0553 Version 1 10