Business Statistics (BK/IBA) Tutorial 1 Exercises

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1 Business Statistics (BK/IBA) Tutorial 1 Exercises Instruction In a tutorial session of 2 hours, we will obviously not be able to discuss all questions. Therefore, the following procedure applies: we expect students to prepare all exercises in advance; we will discuss only a selection of exercises; exercises that were not discussed during class are nevertheless part of the course; students can indicate their wish list of exercises to be discussed during the session; teachers may invite students to answer questions, orally or on the blackboard. We further understand that your time is limited, and in particular that your time between lecture and tutorial may be limited. In case you have no time to prepare everything, we kindly advise you to give priority to the exercises that are indicated with the are not relevant! icon. This does not mean that the other questions 1A Data Q1 (Doane & Seward, 4/E, 2.1) What type of data (categorical, discrete numerical, or continuous numerical) is each of the following variables? If there is any ambiguity about the data type, explain why the answer is unclear. a. The manufacturer of your car. b. Your college major. c. The number of college credits you are taking. Q2 (Doane & Seward, 4/E, 2.2) What type of data (categorical, discrete numerical, or continuous numerical) is each of the following variables? If there is any ambiguity, explain why the answer is unclear. a. Length of a TV commercial. b. Number of peanuts in a can of Planter s Mixed Nuts. c. Occupation of a mortgage applicant. d. Flight time from London Heathrow to Chicago O Hare. Q3 (Doane & Seward, 4/E, 2.3) What type of data (categorical, discrete numerical, or continuous numerical) is each of the following variables? If there is any ambiguity about the data type, explain why the answer is unclear. a. The miles on your car s odometer. b. The fat grams you ate for lunch yesterday. c. The name of the airline with the cheapest fare from New York to London. d. The brand of cell phone you own. Q3 a. Continuous numerical (often represented as discrete numerical) b. Continuous numerical (often reported as an integer); c. Categorical; d. Categorical Q2 a. Continuous numerical; b. Discrete numerical; c. Categorical; d. Continuous numerical Q1 a. Categorical; b. Categorical; c. Discrete numerical Q4 (Doane & Seward, 4/E, 2.9) BS 1 Tutorial 1

2 Which measurement level (nominal, ordinal, interval, ratio) is each of the following variables? Explain. a. Number of hits in Game 1 of the next World Series. b. Baltimore s standing in the American League East (among five teams). c. Field position of a baseball player (catcher, pitcher, etc.). d. Temperature on opening day (Celsius). e. Salary of a randomly chosen American League pitcher. f. Freeway traffic on opening day (light, medium, heavy). Q5 (Doane & Seward, 4/E, 2.10) Which measurement level (nominal, ordinal, interval, ratio) is each of the following variables? Explain. a. Number of employees in the Walmart store in Hutchinson, Kansas. b. Number of merchandise returns on a randomly chosen Monday at a Walmart store. c. Temperature (in Fahrenheit) in the ice-cream freezer at a Walmart store. d. Name of the cashier at register 3 in a Walmart store. e. Manager s rating of the cashier at register 3 in a Walmart store. f. Social security number of the cashier at register 3 in a Walmart store. 1B Q1 Summarizing data (based on Doane & Seward, 4/E, 4.10.a) Given is the following data set with exam scores (9 students) 42, 55, 65, 67, 68, 75, 76, 78, 94. a. Find the median, midrange, and geometric mean. You may use your calculator. b. Are they reasonable measures of central tendency? Explain. Q2 (Doane & Seward, 4/E, 4.18) The number of Internet users in Latin America grew from 78.5 million in 2000 to million in Use the geometric mean to find the mean annual growth rate. Source: (Accessed April 5, 2011). Q3 (Doane & Seward, 4/E, 4.20) For each data set A: 6, 7, 8; B: 4, 5, 6, 7, 8, 9, 10; C: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 a. Find the mean. b. Find the standard deviation, treating the data as a sample. c. Find the standard deviation, treating the data as a population. d. What does this exercise show about the two formulas? Q3 Q2 7.15% Q1 a. 68, 68, Q5 a. Ratio; b. Ratio; c. Interval; d. Nominal; e. Ordinal; f. Nominal Q4 a. Ratio; b. Ordinal; c. Nominal; d. Interval; e. Ratio; f. Ordinal BS 2 Tutorial 1

3 Q4 (Doane & Seward, 4/E, 4.23) Given are data summaries on three stocks. A: x = $24.50, s = 5.25; B: x = $147.25, s = 12.25; C: x = $5.75, s = 2.08 a. Find the coefficient of variation for prices of these three stocks. b. Which stock has the greatest relative variation? c. To measure variability, why not just compare the standard deviations? Q5 (Doane & Seward, 4/E, 4.34) Scores on an accounting exam ranged from 42 to 96, with quartiles Q 1 = 61, Q 2 = 77, and Q 3 = 85. a. Sketch a simple boxplot (5 number summary without fences) using a nicely scaled X-axis. b. Describe its shape (skewed left, symmetric, skewed right). Q6 (Doane & Seward, 4/E, 4.40) For each X-Y data set (n = 12): a. Make a scatter plot. (You may use Excel or SPSS) b. Find the sample correlation coefficient. (You may use your calculator) c. Is there a linear relationship between X and Y? If so, describe it. Note: Use Excel or MegaStat or MINITAB or SPSS. See XYDataSets Q6 b. ra = ; rb = ; rc = Q5 a. b. The long left whisker suggests left-skewness. Q4 ; b. Stock C 2A Basic Probability Q1 (Doane & Seward, 4/E, 5.13) Are these characteristics of a student at your university mutually exclusive or not? Explain. a. A = works 20 hours or more, B = majoring in accounting b. A = born in the United States, B = born in Canada c. A = owns a Toyota, B = owns a Honda BS 3 Tutorial 1

4 Q2 (Doane & Seward, 4/E, 5.15) Given P(A) =.40, P(B) =.50, and P(A B) =.05, find a. P(A B), b. P(A B), and c. P(B A). (d) Sketch a Venn diagram. Q3 (Doane & Seward, 4/E, 5.17) Suppose Samsung ships 21.7 percent of the liquid crystal displays (LCDs) in the world. Let S be the event that a randomly selected LCD was made by Samsung. Find a. P(S) b. P(S ) c. the odds in favor of event S d. the odds against event S (Data are from The Economist 372, no [July 24, 2004], p. 59.) Q4 (Doane & Seward, 4/E, 5.19) List two binary events that describe the possible outcomes of each situation. a. A pharmaceutical firm seeks FDA approval for a new drug. b. A baseball batter goes to bat. c. A woman has a mammogram test. Q5 (based on Doane & Seward, 4/E, 5.21) Let S be the event that a randomly chosen female aged is a smoker. Let C be the event that a randomly chosen female aged is a Caucasian. Given P(S) =.246, P(C) =.830, and P(S C) =.232, find each probability and express the event in words. (Data are from Statistical Abstract of the United States, 2001.) a. Make a contingency table with the data available and complete the table. Use this to find b. P(S ). c. P(S C). d. P(S C). e. P(S C ). f. Are C and S independent? Q6 (Doane & Seward, 4/E, 5.23) Given P(A) =.40, P(B) =.50, and P(A B) =.05. a. Find P(A B). b. In this problem, are A and B independent? Explain. Q6 a. 0.10; b. No Q5 b ; c ; d ; e ; f. No Q4 a. X = 1 if the drug is approved, 0 otherwise. b. X = 1 if batter gets a hit, 0 otherwise. c. X = 1 if breast cancer detected, 0 otherwise. Q3 a ; b ; c ; d Q2 a. 0.85; b. 0.10; c Q1 a. Not mutually exclusive. b. Mutually exclusive. c. Not mutually exclusive. BS 4 Tutorial 1

5 Q7 (Doane & Seward, 4/E, 5.25) The probability that a student has a Visa card (event V) is. 73. The probability that a student has a MasterCard (event M) is. 18. The probability that a student has both cards is. 03. a. Find the probability that a student has either a Visa card or a MasterCard (or both). b. In this problem, are V and M independent? Explain. Q8 P(A) = 0.4, P(B) = 0.6, A and B are independent. Find P(A B). Q9 (Doane & Seward, 4/E, 5.30) The contingency table below shows the results of a survey of online video viewing by age. Find the following probabilities or percentages: a. Probability that a viewer is aged b. Probability that a viewer prefers watching TV videos. c. Percentage of viewers who are and prefer watching user created videos. d. Percentage of viewers aged who prefer watching user created videos. e. Percentage of viewers who are or prefer user created videos? Q10 (Doane & Seward, 4/E, 5.36) The following contingency table shows average yield (rows) and average duration (columns) for 38 bond funds. For a randomly chosen bond fund, find the probability that: a. The bond fund is long duration. b. The bond fund has high yield. c. The bond fund has high yield given that it is of short duration. d. The bond fund is of short duration given that it has high yield. Q10 a ; b ; c ; d Q9 a. 0.69; b. 0.48; c. 0.39; d ; e Q Q7 a. 0.88; b. No 2B Probability distributions Q1 (Doane & Seward, 4/E, 6.3) BS 5 Tutorial 1

6 On the midnight shift, the number of patients with head trauma in an emergency room has the probability distribution shown below. a. Calculate the mean and standard deviation. b. Describe the shape of this distribution. Q2 The following bivariate distribution is given: a. Find μ X and μ Y b. Find σ X 2, σ X, σ Y 2, and σ Y, c. Find the distribution of X + Y. d. Find μ X+Y. Q3 (Doane & Seward, 4/E, 6.9) The ages of Java programmers at SynFlex Corp. range from 20 to 60. a. If their ages are uniformly distributed, what would be the mean and standard deviation? b. What is the probability that a randomly selected programmer s age is at least 40? At least 30? Hint: Treat employee ages as integers. Q4 (Doane & Seward, 4/E, 6.15.b) Find the mean and standard deviation for a binomial random variable with n = 10 and π =.40. Q5 (Doane & Seward, 4/E, 6.18.b) Calculate the binomial probability of X = 1 with n = 10, π =.40. Q6 (Doane & Seward, 4/E, 6.18.c) Calculate the binomial probability of X = 3 with n = 12, π =.70. Q6 P(X = 3) = Q5 P(X = 1) = Q4 μ = 4, σ = Q3 a. μ = 40 and σ = 11.83; b. P(X 40) = and P(X 30) = Q2 a. E(X) = 170; E(Y) = 100; b. σx = 45.83; σy = c. P(X + Y = 150) = 0.2, P(X + Y = 250) = 0.4, P(X + Y = 350) = 0.4. d. E(X + Y) = 270. Q1 a and b. The distribution is skewed to the right. Q7 (based on Doane & Seward, 4/E, 6.19) If X has a binomial distribution, calculate each compound event probability, as well as μ and σ 2 : a. X 3, n = 8, π =.20. BS 6 Tutorial 1

7 b. X > 7, n = 10, π =.50. c. X < 3, n = 6, π =.70. d. X 10, n = 14, π =.95. Q8 (Doane & Seward, 4/E, 7.11) State the Empirical Rule for a normal distribution (see Chapter 4). Q9 (Doane & Seward, 4/E, 7.13) Find the standard normal area for each of the following, showing your reasoning clearly and indicating which table you used. a. P(0 < Z < 0.50). b. P( 0.50 < Z < 0). c. P(Z > 0). d. P(Z = 0) Q10 (Doane & Seward, 4/E, 7.16) Find the standard normal area for each of the following. Sketch the normal curve and shade in the area represented below. a. P(Z < 1.96). b. P(Z > 1.96). c. P(Z < 1.65). d. P(Z > 1.65). Q11 (Doane & Seward, 4/E, 7.21) Find the associated z-score for each of the following standard normal areas. a. Lowest 6 percent b. Highest 40 percent c. Lowest 7 percent Q12 (Doane & Seward, 4/E, 7.27) Assume that the number of calories in a McDonald s Egg McMuffin is a normally distributed random variable with a mean of 290 calories and a standard deviation of 14 calories. a. What is the probability that a particular serving contains fewer than 300 calories? b. More than 250 calories? c. Between 275 and 310 calories? Show all work clearly. (Data are from McDonalds.com) Q12 a ; b ; c Q11 a. z = 1.555; b. z = 0.25; c. z = 1.48 Q10 a ; b ; c ; d Q9 a ; b ; c ; d. 0 Q8 It says that for data from a normal distribution we expect * about 68.26% will lie within μ ± 1σ * about 95.44% will lie within μ ± 2σ * about 99.73% will lie within μ ± 3σ Q7 a. P(X 3) = ; μ = 1.60; σ 2 = b. P(X > 7) = ; μ = 5.0; σ 2 = 2.5. c. P(X < 3) = ; μ = 4.2; σ 2 = d. P(X 10) = ; μ = 13.3; σ 2 = BS 7 Tutorial 1

8 Q13 (Doane & Seward, 4/E, 7.37) The weight of newborn babies in Foxboro Hospital is normally distributed with a mean of 6.9 pounds and a standard deviation of 1.2 pounds. a. How unusual is a baby weighing 8.0 pounds or more? b. What would be the 90th percentile for birth weight? c. Within what range would the middle 95 percent of birth weights lie? Q14 (based on Doane & Seward, 4/E, 7.5) Find each uniform continuous probability and sketch a graph showing it as a shaded area. Also find E(X) and var(x) for each case. a. P(X < 10) for U(0,50) b. P(X > 500) for U(0, 1,000) c. P(25 < X < 45) for U(15, 65) Q15 (Doane & Seward, 4/E, 7.8) Assume the weight of a randomly chosen American passenger car is a uniformly distributed random variable ranging from 2,500 pounds to 4,500 pounds. a. What is the mean weight of a randomly chosen vehicle? b. The standard deviation? c. What is the probability that a vehicle will weigh less than 3,000 pounds? d. More than 4,000 pounds? e. Between 3,000 and 4,000 pounds? Q16 (based on Doane & Seward, 4/E, 7.29) The pediatrics unit at Carver Hospital has 24 beds. The number of patients needing a bed at any point in time is N(19.2,2.5). a. What is the probability that the number of patients needing a bed will exceed the pediatric unit s bed capacity? b. Could you really apply the normal distribution? Q17 X and Y are two random variables, X is the return of 100 shares of stock A, Y is the return of 100 shares of stock B. It is known that both X and Y are normally distributed with mean 3 and standard deviation 5 (dollars). The covariance between X and Y is σ X,Y = 20. a. Find the probability P(2X 10) b. Find the probability P(X + Y 10) (assume that X + Y is normally distributed) Q17 a ; b Q16 a Q15 a. 3500; b ; c. 0.25; d. 0.25; e Q14 a. P(X 10) = 0.2; μx = 25; σx 2 = b. P(X > 500) = 05; μx = 500; σx 2 = c. P(25 < X < 45) = 0.4; μx = 40; σx 2 = Q13 a pounds; b pounds; c. between 4.5 and 9.3 pounds Q18 (Doane & Seward, 4/E, 7.49) The probability that a vending machine in the Oxnard University Student Center will dispense the desired item when correct change is inserted is.90. If 200 customers try the machine, find the probability that a. at least 175 will receive the desired item b. that fewer than 190 will receive the desired item BS 8 Tutorial 1

9 Q19 In a large population 40% of the people travel by train. Approximate the probability using an appropriate approximating distribution - that in a random sample of size n = 20 a proportion of 0.50 or less travels by train. Old exam questions Q1 23 March 2016, Q1c Grades for the marketing exam have a right-skewed distribution, with μ = 5.0 and σ = 1.0. In total, 289 students take the exam. What is the probability that a randomly selected student has a score of at least the 95 percentile or higher? Note that the answer may be not enough information. (text or 2 decimals) Q2 23 March 2016, Q1h We roll a die 2 times, and indicate the results as X 1 and X 2. Find P(X 1 2 X 1 + X 2 = 5). (2 decimals) Q Q Q Q18 a ; b BS 9 Tutorial 1