Descriptive Statistics - Numerical Measures Part 5

121. The following observations are given for two variables. x 5 8 18 20 22 30 10 7 a. b. y 2 12 3 6 11 19 18 9 Compute and interpret the sample covariance for the above data. Compute and interpret the sample correlation coefficient. ANSWER: 19.3. Since the covariance is positive, a positive relationship between x and y is indicated. 0.345. There is a positive relationship between x and y. The relationship is not very b. strong. a. 122. The following data represent the daily demand (y in thousands of units) and the unit price (x in dollars) for a product. Daily Demand (y) 47 39 35 44 34 20 15 30 a. b. Unit Price (x) 1 3 5 3 6 8 16 6 Compute and interpret the sample covariance for the above data. Compute and interpret the sample correlation coefficient. ANSWER: a. –47.00 (rounded). Since the covariance is negative, a negative relationship between x and y is indicated. b. –0.922. There is a strong negative relationship between daily demand and unit price. 123. Compute the weighted mean for the following data. xi 9 8 5 3 2 Weight (wi) 10 12 4 5 3 6.68 ANSWER: 124. Compute the weighted mean for the following data. xi 19 17 14 13 18 ANSWER: Weight (wi) 12 30 28 10 10 16 125. Paul, a freshman at a local college, just completed 15 credit hours. His grade report is presented below. Course Credit Hours Grade Calculus 5 C Biology 4 A English 3 D Music 2 B P.E. 1 A The local university uses a 4-point grading system, i.e., A = 4, B = 3, C = 2, D = 1, F = 0. Compute Paul's semester grade point average. 2.6 ANSWER: 126. Suppose the mean GMAT score is 550 with a standard deviation of 100. Hayden takes the GMAT and is told his zscore is 2.5. Interpret the meaning of a z-score in this context. Since Hayden’s z-score is positive, he scored above the mean of 550. The value, 2.5, indicates he scored 2.5 standard deviations from the mean. One standard deviation is 100, so 2.5 standard deviations is 250. Therefore, Hayden scored 250 points above the mean of 550, or 800 on the GMAT. ANSWER: 127. Explain why Chebyshev’s theorem cannot be used for one standard deviation from the mean. Chebyshev’s Theorem requires an input value of at least 2, since an input of 1 would cause a divide by 0 error. ANSWER: 128. Use this data set to answer the following questions. 15 68 11 26 53 15 14 13 36 20 a. Find the five-number summary. b. Find the IQR. c. Use IQR to determine whether there are any outliers. Explain. a. 11, 14, 17.5, 36, 68 ANSWER: b. 22 c. 68 is a suspected outlier, since it seems to be much higher than most other data values. 1.5(IQR) = 33, and 33 + 36 = 69. Since 68 < 69, it is not an outlier. 129. Use Excel to make a box plot of the following data: 15 68 11 26 53 15 14 13 36 20 ANSWER: 130. Use Excel to make a box plot of the following data: 115 184 191 216 153 185 194 183 236 202 ANSWER: 131. Use this data set to answer the following questions. 115 184 a. b. c. 191 216 153 185 194 183 236 202 Find the five-number summary. Find the IQR. Use IQR to determine whether there are any outliers. Explain. ANSWER: a. 115, 183, 188, 202, 236 b. 19 c. 1.5(IQR) = 28.5. 183 – 28.5 = 154.5 and 202 + 28.5 = 230.5 115 and 153 are both less than 154.5, and 236 is greater than 230.5, so 115, 153, and 236 are all outliers. 132. Describe how a comparative analysis using box plots of men’s vs. women’s salaries at ACME, Inc. could be used. Two box plots, one depicting men’s and one depicting women’s salaries, side by side would visually show differences between minimum, maximum, and median values. Differences or similarities of variability could also be easily recognized, as would any outliers in either direction. ACME, Inc. could use this depiction to correct gender bias, if it existed, with regard to salary, or the company could publish the comparative analysis using box plots to demonstrate no gender bias exists, if it doesn’t. ANSWER: 133. Del Michaels had a successful morning, or so he thinks, selling 1,300 surplus notebook computers over the telephone to three commercial customers. The three customers were not equally skillful at negotiating a low unit price. Customer A bought 600 computers for $1,252 each, B bought 300 units at $,1,310 each, and C bought 400 at $1,375 each. a. What is the average unit price at which Del sold the 1,300 computers? b. Del’s manager told Del he expected him to sell, by the end of the day, a total of 2,500 surplus computers at an average price of $1,312 each. What is the average unit price at which Del must sell the remaining 1,200 computers? a. $1,303.23 ANSWER: b. $1,321.50 134. Missy Walters owns a mail-order business specializing in baby clothes. She is considering offering her customers a discount on shipping charges based on the dollar amount of the mail order. Before Missy decides the discount policy, she needs a better understanding of the dollar amount distribution of the mail orders she receives. Missy had an assistant randomly select 50 recent orders and record the value, to the nearest dollar, of each order as shown below. 136 211 194 277 231 281 162 242 348 154 226 212 368 173 166 123 241 258 409 214 178 182 323 264 311 445 290 196 237 141 231 434 183 490 159 389 167 209 222 362 196 246 198 472 189 175 338 212 248 260 a. Determine the mean, median, and mode for this data set. b. Determine the 80th percentile. c. Determine the first quartile. d. Determine the range and interquartile range. e. Determine the sample variance, sample standard deviation, and coefficient of variation. f. Determine the z-scores for the minimum and maximum values in the data set. ANSWER: a. mean = 251.46, median = 228.5, modes = 196 and 231 b. 331.5 c. 183 d. range = 367, interquartile range = 107 e. variance = 8398.5, standard deviation = 91.64, coefficient of variation = 36.44 f. minimum’s z-score = –.40, maximum’s z-score = 2.60 135. Ron Butler, a custom home builder, is looking over the expenses he incurred for a house he just finished constructing. To price future construction projects, he would like to know the average wage ($/hr.) he paid the workers he employed. Listed below are the categories of worker he employed, along with their respective wage and total hours worked. What is the average wage ($/hr.) he paid the workers? Worker Carpenter Electrician Laborer Painter Plumber ANSWER: Wage ($/hr.) 21.60 28.72 11.80 19.75 24.16 Total Hours 520 230 410 270 160 $20.05 136. Interpret the scatter plot in terms of correlation. ANSWER: There is a moderate, positive linear correlation between the ages of spouses at first marriage. 137. Interpret the scatter plot in terms of correlation. ANSWER: There is a very strong, negative linear correlation between the age of a car and its value. 138. Angela Lopez, a golf instructor, is interested in investigating the relationship between a golfer’s average driving distance and 18-hole score. She recently observed the performance of six golfers during one round of a tournament and measured, as accurately as possible, the distances (yards) of their drives and noted their final scores. She then computed each golfer’s average drive distance for 18 holes. The results of her sample are shown below. Golfer 1 2 3 4 Avg. Drive (yards) 277.6 259.5 269.1 267.0 18-Hole Score 69 71 70 70 5 6 255.6 272.9 71 69 Compute and interpret both the sample covariance and the sample correlation coefficient. ANSWER: The sample covariance value of –7.08 indicates a negative relationship between driving distance and final score. This is not surprising. Unless a golfer’s putting performance is relatively poor, we would expect longer driving distances to lead to lower final scores. The correlation coefficient value of –.96 indicates (because it is so close to –1.00) an exceptionally strong negative correlation between driving distance and final score. 139. Reed Auto periodically has a special weeklong sale. As part of the advertising campaign, Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of five previous sales are shown below. Week TV Ads Cars Sold 1 1 14 2 3 24 3 2 18 4 1 17 5 3 27 Compute and interpret both the sample covariance and the sample correlation coefficient. ANSWER: Sample covariance = 5.0; sample correlation coefficient = .937