Business Statistics (BK/IBA) Tutorial 1 Full solutions

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1 Business Statistics (BK/IBA) Tutorial 1 Full solutions 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!!! sign. 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. A1 a. Categorical; b. Categorical; c. Discrete numerical!!! 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. A2 a. Continuous numerical; b. Discrete numerical; c. Categorical; d. Continuous numerical 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. A3 a. Continuous numerical (often represented as discrete numerical) b. Continuous numerical (often reported as an integer); c. Categorical; d. Categorical!!! Q4 (Doane & Seward, 4/E, 2.9) Which measurement level (nominal, ordinal, interval, ratio) is each of the following variables? Explain. BS 1 Tutorial 1

2 A4 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). a. Ratio; b. Ordinal; c. Nominal; d. Interval; e. Ratio; f. Ordinal 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. A5 Sol a. Ratio; b. Ratio; c. Interval; d. Nominal; e. Ordinal; f. Nominal a. Ratio. The number of employees is a count and you can have zero employees. b. Ratio. The number of returns is a count and you can have zero returns. c. Interval. The temperature difference from 70 degrees to 80 degrees is the same increase as 80 degrees to 90 degrees. However, zero temperature does not mean no temperature exists, therefore it is interval. d. Nominal. It is not a number and you could not rank order this cashier with others. e. Ordinal. Ratings of employees generally fall into categories such as exceeds standards, etc. Therefore, we know it is either nominal or ordinal and since we can rank order this employee with others given their rankings, we can say it is ordinal. f. Nominal. There is no meaningful zero and distance between social security numbers has no meaning. We also would not rank order based on social security number so this is nominal even though it is a number. Summarizing data!!! Q1 (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. A1 a. 68, 68, Sol a. Median = 68, Midrange = 68, Geometric Mean = ( ) b. The midrange is not very robust because it is sensitive to extreme data such as the 94 for one exam score. The median and geometric mean appear to be close in value. However, we might want to rely on the median because the data do not appear to have a steep increasing trend and the median is a more easily understood and recognized measure of central tendency. 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). A2 7.15% BS 2 Tutorial 1

3 Sol Extra Note the use of 1 because there are growth rates.!!! 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? A3 a-c. Sol d. The sample standard deviation is larger than the population standard deviation for the same data set. This makes sense because in the formula for sample standard deviation we divide by 1 instead of to correct for error. Dividing by 1 instead of means dividing by a smaller number which means getting a larger number as a result. This exercise shows us that samples can have similar means, but different standard deviations. We cannot get a sense of what the standard deviation is just by looking at the mean. Q4 (Doane & Seward, 4/E, 4.23) Given are data summaries on three stocks. A: 24.50, 5.25; B: , 12.25; C: 5.75, 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? A4 a. Sol b. Stock C b. Stock C, the one with the smallest standard deviation and smallest mean, has the greatest relative variation. c. The stocks have different average values therefore directly comparing the standard deviations is not a good comparison of risk. The variation relative to the mean value is more appropriate. Q5 (Doane & Seward, 4/E, 4.34) Scores on an accounting exam ranged from 42 to 96, with quartiles " 61, " 77, and " % 85. a. Sketch a simple boxplot (5 number summary without fences) using a nicely scaled &-axis. b. Describe its shape (skewed left, symmetric, skewed right). A5 a. BS 3 Tutorial 1

4 b. The long left whisker suggests left-skewness. Q6 (Doane & Seward, 4/E, 4.40) For each &-' data set ( 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 A6 b. ( ) ; ( * ; ( Sol a. To calculate the sample correlation coefficient we can use the formula ( /4-. / / 0312 /4-. / /4-3 / or use the =CORREL(XData, Y Data) function in Excel, or use the correlation menu from SPSS. This gives us ( There appears to be a strong, negative, linear relationship. b. BS 4 Tutorial 1

5 Again we can use the equation or Excel or SPSS and the result is ( There appears to be a strong, positive, linear relationship. c. Again, we can use the formula or Excel or SPSS and the result is ( There appears to be a weak, positive, linear relationship. 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 A1 a. Not mutually exclusive. b. Mutually exclusive. BS 5 Tutorial 1

6 Sol c. Not mutually exclusive. a. Not mutually exclusive, you can both work 20 hours or more and be an accounting major. b. Mutually exclusive, you cannot be born in Canada and in the United States. You have to be one or the other. c. Not mutually exclusive, you can own two different cars.!!! Q2 (Doane & Seward, 4/E, 5.15) Given , , and , find a , b , and c (d) Sketch a Venn diagram. A2 a. 0.85; b. 0.10; c Sol a < b =->?2 =-?2 c =-? >2 =->2 d. @.A@ Q3 (Doane & Seward, 4/E, 5.17) Suppose Samsung ships 21.7 percent of the liquid crystal displays (LCDs) in the world. Let B be the event that a randomly selected LCD was made by Samsung. Find a. 6-B2 b. 6-B C 2 c. the odds in favor of event B d. the odds against event B (Data are from The Economist 372, no [July 24, 2004], p. 59.) A3 a ; b ; c ; d Sol a. 6-B b. 6-B C 2 16-B c. Odds in favor d. Odds 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. A4 a. & 1 if the drug is approved, 0 otherwise. b. & 1 if batter gets a hit, 0 otherwise. BS 6 Tutorial 1

7 c. & 1 if breast cancer detected, 0 otherwise.!!! Q5 (based on Doane & Seward, 4/E, 5.21) Let B be the event that a randomly chosen female aged is a smoker. Let E be the event that a randomly chosen female aged is a Caucasian. Given 6-B2.246, 6-E2.830, and 6-B E2.232, 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. 6-B C 2. c. 6-B E2. d. 6-B E2. e. 6-B E C 2. f. Are E and B independent? A5 b ; c ; d ; e ; f. No Sol a. Blue is given, then finish the table b. 6-B C 2 16-B There is a 75.4% chance that a female aged is a nonsmoker. c. 6-B E < There is an 84.4% chance that a female aged is a smoker or is Caucasian. d. 6-B Given that the female aged is a Caucasian, there is a chance that she is a smoker. e. 6-B E C 2 6-B26-B E B E C that the female aged is not Caucasian, there is an 8.24% chance that she is a smoker. f. In case of independence, cell probabilities equal product of row and column probabilities. (You can stop verifying as soon as you have found an exception) Q6 (Doane & Seward, 4/E, 5.23) Given , , and a. Find b. In this problem, are 7 and 8 independent? Explain. A6 Sol a. 0.10; b. No a b. No, 7 and 8 are not independent because F 6-72 or because F Q7 (Doane & Seward, 4/E, 5.25) The probability that a student has a Visa card (event G) is.73. The probability that a student has a MasterCard (event H) is.18. The probability that a student has both cards is.03. BS 7 Tutorial 1

8 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. A7 a. 0.88; b. No Sol a. 6-G H2 0.73< b. 6-G H2 F 6-G26-H2; therefore G and H are not independent. Q , , 7 and 8 are independent. Find A Sol !!! 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? A9 a. 0.69; b. 0.48; c. 0.39; d ; e Sol a. 6-aged N b. 6-prefers TV videos2 A c. 6-aged 1834 and prefers user created videos2 %N d. 6-prefers user created videos aged %N N e. 6-aged 3554 prefers user created videos2 6-aged 35542<6-prefers user created videos2 6-aged 3554 prefers user created videos2 0.20< 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: BS 8 Tutorial 1

9 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. A10 a ; b ; c ; d Sol a. 6-\ % 2 % b. 6-' % 2 % c. 6-' % \ 2 =-3 ] ^2 =-^2 /% Another way is because the two 38s /% cancel each other. d. 6-\ ' % 2 =-^ 3 ] 2 /% Another way is because the two 38s =-3 ] 2 /% cancel each other. Probability distributions Q1 (Doane & Seward, 4/E, 6.3) 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. A1 a and b. The distribution is skewed to the right. Sol `-&2 a `-&2b Total a. `-&2 2.25, G-& , c b. The distribution is skewed to the right (make graph). BS 9 Tutorial 1

10 !!! Q2 The following bivariate distribution is given: a. Find d. and d 3 b. Find c., c., c 3, and c 3, c. Find the distribution of & <'. d. Find d.e3. A2 a. `-&2 170; `-'2 100; b. c ; c c. 6-& <' , 6-& <' , 6-& <' d. `-& <' Sol a. a. `-& < ; `-' < (or use symmetry!) b. c. f < c 3 f < c. 6-& <' , 6-& <' , 6-& <' d. `-& <'2 `-&2<`-'2 170< !!! 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. A3 a. d 40 and c 11.83; b. 6-& g and 6-& g Sol a. With h 20 and i 60, 40 and c -@0@e b. 6-& g & j %N0@e and 6-& g 302 BS 10 Tutorial 1

11 Q4 (Doane & Seward, 4/E, 6.15.b) Find the mean and standard deviation for a binomial random variable with 10 and k.40. A4 d 4, c Sol d , c f Q5 (Doane & Seward, 4/E, 6.18.b) Calculate the binomial probability of & 1 with 10, k.40. A5 6-& Sol 6-& Extra Use the table if possible, do not use the graphical calculator or Excel. Q6 (Doane & Seward, 4/E, 6.18.c) Calculate the binomial probability of & 3 with 12, k.70. A6 6-& Sol 6-& 32! %!-0%2! % % Extra Use the table if possible, do not use the graphical calculator or Excel.!!! Q7 (based on Doane & Seward, 4/E, 6.19) If & has a binomial distribution, calculate each compound event probability, as well as d and c : a. & j 3, 8, k.20. b. & m 7, 10, k.50. c. & n 3, 6, k.70. d. & j 10, 14, k.95. A7 a. 6-& j ; d 1.60; c b. 6-& m ; d 5.0; c 2.5. c. 6-& n ; d 4.2; c d. 6-& j ; d 13.3; c Sol a. 6-& j <0.3355<0.2936< d ; c b. 6-& m <0.0098< d ; c c. 6-& n 32 6-& j <0.0102< d ; c d. 6-& j < d ; c Extra With Excel, use binomdist(3,8,.2,1) = etc. Q8 (Doane & Seward, 4/E, 7.11) State the Empirical Rule for a normal distribution (see Chapter 4). A8 It says that for data from a normal distribution we expect * about 68.26% will lie within d o1c * about 95.44% will lie within d o2c * about 99.73% will lie within d o3c Extra It is convenient to remember the number 68, 95, and 99.7.!!! 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. BS 11 Tutorial 1

12 a. 6-0 n p n b n p n 02. c. 6-p m 02. d. 6-p 02 A9 a ; b ; c ; d. 0 Sol a. 6-0 n p n p n p j b Use a graph and compare with a) c d. 0. Extra It is important to be able to do this without a graphical calculator!!!! 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. 6-p n b. 6-p m c. 6-p n d. 6-p m A10 a ; b ; c ; d Sol a. Using the table in Appendix C-2, Extra You can also use the 5% from the empirical rule, and divide by 2 because you need only one tail. b. 6-p m p n Using Appendix C-2: 16-p n We also know that the area under the curve for the lower tail at 1.96 will be the same as the area under the curve for the upper tail at <1.96 so we could have just used part a) to get the answer. c. Using the table in Appendix C-2, 6-p n d. 6-p m p n It is important to be able to do this without a graphical calculator!!!! Q11 (Doane & Seward, 4/E, 7.21) Find the associated q-score for each of the following standard normal areas. a. Lowest 6 percent b. Highest 40 percent c. Lowest 7 percent A11 a. q 1.555; b. q 0.25; c. q 1.48 Sol Using Appendix C-2: a. q (the area 0.06 is halfway between and ). Excel: q b. q 0.25 (the closest area to 0.6 is ). Excel: q c. q 1.48 (closest area is ). Excel: q Extra It is important to be able to do this without a graphical calculator!!!! Q12 (Doane & Seward, 4/E, 7.27) BS 12 Tutorial 1

13 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) A12 a ; b ; c Sol a. 6-& n r.0s u 6-p n Extra b. 6-& m r.0s n %@@0N@ t A t A c n & n r 0N@ A 6-p n p j u 6-p m p j n.0s n %@0N@ t A u n p n Do not use Excel (NORMDIST(300,290,14,TRUE) = ) or a graphical calculator, because you are not allowed to use these tools at the exam! 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? A13 a pounds; b pounds; c. between 4.5 and 9.3 pounds Sol a. 6-& g 82 6r.0s g 0.N u 6-p g t. This probability indicates that the event is not common but not unlikely. One could also say that an 8 pound baby is at approximately the 80th percentile which also indicates a pretty high weight. b. 6-& j 2 6r.0s 0.N u pounds. Extra j 0.N t. c. 6-& j 2 6rp j 50.N u N.. and 6-& j 2 6rp j 0.N u N pounds pounds Please always use standardizing! With Excel, it is a. 1-NORMDIST(7,6.9,1.2,TRUE) = b. NORMINV(0.9,6.9,1.2) = c. NORMINV(0.025,6.9,1.2) = and NORMINV(.975,6.9,1.2) = 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 `-&2 and var-&2 for each case. a. 6-& n 102 for w-0,502 b. 6-& m 5002 for w-0,1,0002 c n & n 452 for w-15,652 A14 a. 6-& j ; d. 25; c b. 6-& m ; d. 500; c c n & n ; d. 40; c Sol a. 6-& j 2 6-& j ; d -0<502 25; c. -@0@ ; c. fc BS 13 Tutorial 1

14 b. 6-& m 2 6-& m ; d -0< ; c. -@@@0@ c. 6-y j & j z2 {0+ A n & n ; d *0) <652 40; c Extra For the uniform distribution d o2c already covers all density mass! 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? A15 a. 3500; b ; c. 0.25; d. 0.25; e Sol a. The equation is: )e* b. The equation is: -*0)25. c -A@@0@@ c. 6-& n %@@@0@@ 0.25 A@@0@@ d. 6-& m A@@0A@@@ 0.25 A@@0@@ e n & n A@@@0%@@@ 0.50 A@@0@@!!! 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 -19.2,2.52. a. What is the probability that the number of patients needing a bed will exceed the pediatric unit s bed capacity? BS 14 Tutorial 1

15 b. Could you really apply the normal distribution? A16 a Sol a. 6-& m 242 6r.0s u 6-p m m A N. t. b. c is only 2.5, and logically only integers are possible. This may be problematic for application of the normal distribution (using at least a continuity correction might solve this problem). The range would be somewhere between d o3c ~11.7,26.7, so the natural barrier of at least 0 patients is not a problem for the normal distribution.!!! Q17 & and ' are two random variables, & is the return of 100 shares of stock 7, ' is the return of 100 shares of stock 8. It is known that both & and ' are normally distributed with mean 3 and standard deviation 5 (dollars). The covariance between & and ' is c.,3 20. a. Find the probability 6-2& j 102 b. Find the probability 6-& <' j 102 (assume that & <' is normally distributed) A17 a ; b Sol a. 6-2& j & j 52 6rp j 0% u 6-p j Extra b. d.e3 `-& <'2 `-&2<`-'2 3<3 6 c.e3 var-&2<var-&2<2cov-&,'2 5 <5 < & <' j 102 6rp u 6-p j Usually returns are in percentages, and usually returns of portfolios are considered. 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 A18 a ; b Sol k g 5 and -1k g 5, so we can use the normal approximation. d 180, c a. 6? ƒ -& g ƒ) -& g b. 6? ƒ -& n ƒ) -& j !!! 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 20 a proportion of 0.50 or less travels by train. A Sol If & is number of persons travelling by train, then &~8-20,0.42. Therefore, 6-Š j * -& j 102 6rp u 6-p j Continuity correction is compulsory. Normal approx OK because k g 5 and -1 k2 g 5 We do not ask for the exact probability (0.8725) BS 15 Tutorial 1