CHAPTER 3 Describing Data: Numerical Basic and Applied Questions QUESTIONS 119 AND 120 ARE BASED ON THE FOLLOWING INFORMATION: The police lieutenant in charge of the traffic division has reviewed the number of traffic citations issued per day by each of the 10 police officers in his division. The data were: 13, 21, 12, 34, 31, 13, 22, 26, 25, and 23. 119. What is the standard deviation for the number of citations issued per day? 120. What is the inter-quartile range for the number of citations issued per day? 138. In spite of its advantage in discounting extreme observations, the median is used less frequently than the mean. Why? 139. Give an example to illustrate that the median should not always be preferred to the mean when the population or sample is skewed. QUESTIONS 143 THROUGH 148 ARE BASED ON THE FOLLOWING INFORMATION: A set of data is mounded, with a mean of 500 and a variance of 576. 143. Approximately what proportion of the observations is greater than 476? 144. Approximately what proportion of the observations is less than 548? 145. Approximately what proportion of the observations is greater than 572? 146. Approximately what proportion of the observations is between 452 and 548? 39 Chapter 3 147. Approximately what proportion of the observations is between 428 and 572? 148. Approximately what proportion of the observations is between 476 and 524? QUESTIONS 149 THROUGH 151 ARE BASED ON THE FOLLOWING INFORMATION: The annual percentage returns on two stocks over a 7-year period were as follows: Stock A: Stock B: 4.01% 6.51% 14.31% 4.41% 19.01% 3.81% -14.69% 6.91% -26.49% 8.01% 8.01% 5.81% 5.81% 5.11% 5.11% 149. Compare the means of these two population distribution. 150. Compare the standard deviations of these two population distributions. 151. Compute an appropriate measure of dispersion for both stocks to measure the risk of these investment opportunities. Which stock is more volatile? QUESTIONS 152 AND 153 ARE BASED ON THE FOLLOWING INFORMATION: The following numbers represent the distance, in miles, that randomly selected ten employees of a firm must travel each way to work from home: 6.5, 14.8, 18.6, 6.5, 17.4, 12.3, 1.9, 12.9, 11.1, and 8.0. 152. Find the mean number of miles driven by the ten employees is 153. The standard deviation of the number of miles driven by the ten employees is QUESTIONS 154 THROUGH 156 ARE BASED ON THE FOLLOWING INFORMATION: In a recent survey, 12 students at a local university were asked approximately how many hours per week they spend on the Internet. Their responses were: 13, 0, 5, 8, 22, 7, 3, 0, 15, 12, 13, and 17. 154. What are the mean and standard deviation for this data? 155. What is the coefficient of variation for this data? 40 Describing Data: Numerical 156. From the data presented above, calculate the inter-quartile range. QUESTIONS 157 THROUGH 159 ARE BASED ON THE FOLLOWING INFORMATION: In a recent survey, 200 top executives were asked how many hours they spend each year in community service. The data are presented below. # of Hours # of Executives 0 but < 20 11 20 but < 40 27 40 but < 60 33 60 but < 80 53 80 but < 100 47 f , f m , and f m x 2 100 but < 120 22 120 but < 140 7 157 Calculate the quantities 158. What is the estimated mean amount of time spent by these executives in community service? 159. What is the estimated standard deviation for this data? 160. Calculate the coefficient of variation for the following sample data: 13.2, 14.7, 17.2, 12.1, 21.8, 8.4, 14.3, 11.0, 9.3, and 8.7 i i i i i . QUESTIONS 161 THROUGH 163 ARE BASED ON THE FOLLOWING INFORMATION: The number of students eating breakfast at the school dining commons was recorded over 110 days last semester. These data are presented below. # of Students # of Days 160 but < 190 11 190 but < 220 27 220 but < 250 42 250 but < 280 23 f , f m , and f m 280 but < 310 7 x . 2 161. Calculate the quantities 162. What is the estimated mean number of students showing up for breakfast? 163. What is the estimated standard deviation for this data? i i i i 41 Chapter 3 QUESTIONS 164 THROUGH 168 ARE BASED ON THE FOLLOWING INFORMATION: A small accounting office is trying to determine its staffing needs for the coming tax season. The manager has collected the following data: 46, 27, 79, 57, 99, 75, 48, 89, and 85. These values represent the number of returns the office completed each year over the past nine years it has been doing tax returns: 164. For this data, what is the mean number of tax returns completed each year? 165. For this data, what is the median number of tax returns completed each year? 166. For this data, what is the variance of the number of tax returns completed each year? 167. For this data, what is the inter-quartile for the number of tax returns completed each year? 168. For this data, what is the coefficient of variation for the number of tax returns completed each year? 169. Compute the mean, standard deviation and inter-quartile range for the following sample data: 12.2, 15.9, 8.1, 19.2, 13.7, 7.2, 12.2, 10.9, 5.4, and 16.8. 170. The manager of 45 sales people examined their monthly expenditures on entertaining clients. He found that the mean amount was $237.50 with a standard deviation of $27.40. Based on this data, would a claim for the amount of $300 be considered unlikely? Why or why not? 171. A researcher interested in determining the average monthly expenditures of college students on DVDs finds that for a sample of 25 students, the mean expenditure was $24.40, and the median expenditure was $21.76. Specify the shape of the histogram for this data. Does this shape make sense? Why? 172. The following values represent the annual snowfall, measured in inches, at a local ski resort over the past 10 years: 123, 87, 143, 133, 182, 176, 96, 104, 201, and 152. What is the mean snowfall over the past 10 years? What is the standard deviation? 42 Describing Data: Numerical 173. The shift supervisor at a local manufacturing plant collected data on the number of defects coming off an assembly line over the past eight weeks (40 work days). She summarized the data in the following frequency table. # of Errors # of Weeks 0 1 1 4 2 7 3 16 4 10 5 2 What is the weighted average number of errors per night? 174. The manager of the help-line at a local software manufacturing firm collected data on the number of calls the help desk received per shift for the last 100 days. The data is summarized in the table below. # of Calls # of Days 80 but < 120 1 120 but < 160 21 160 but < 200 29 200 but < 240 32 240 but < 280 15 280 but < 320 2 What is the estimated mean number of calls per night? What is the estimated standard deviation? QUESTIONS 175 THROUGH 178 ARE BASED ON THE FOLLOWING INFORMATION: An investment councilor recently reviewed the account activity of a sample of 10 of his clients and calculated the average number of stock trades per month over the past year for each client. He obtained the following data values: 10.2, 2.5, 11.4, 3.2, 1.1, 3.4, 8.4, 9.7, 11.2, and 2.4. 175. Calculate the average number of trades per month for these 10 clients. 176. Calculate the standard deviation. 177. Calculate the median number of trades per month for these 10 clients. 178. Describe the shape of the distribution of the number of trades per month for these 10 clients. Justify your answer. 180. The supervisor of a tourist information desk at a local airport is interested in how long it takes an employee to serve a customer. Using a stopwatch, he measures the amount of time it takes for each of 10 customers. These times, measured in minutes, are reported as follows: 2.3, 1.5, 3.9, 0.6, 2.7, 3.1, 2.8, 0.9, 1.4, and 2.6. Calculate the standard deviation and the inter-quartile range. 43 Chapter 3 181. Consider the following sample data: 153, 178, 203, 410, 310, 231, 190, and 225. Compute the mean and median. Is the distribution of these numbers skewed to the right, skewed to the left or symmetric? Why? 182. For data that has a mound-shaped distribution, will the inter-quartile range span a larger set of values than the range from one standard deviation below the mean to one standard deviation above the mean? Explain why or why not. 183. For the sample data shown below, compute the five-number summary, and sketch out the Box-and-Whisker plot. 34 49 76 62 29 34 59 92 71 87 43 23 24 64 92 37 82 54 79 27 184. Since 1985, the daily temperature in Daytona Beach, Florida during spring break week has a mean of 74.6 degrees and a standard deviation of 2.2 degrees. Assuming that the distribution of temperatures is mound-shaped, how likely is it that the average daily temperature this year will be above 71 degrees? Explain. 185. Suppose that you are working with a data set and want to check for any outliers. What should you do? Suppose you detect an outlier. What are some of your options, and how would you make your decision? QUESTIONS 194 THROUGH 200 ARE BASED ON THE FOLLOWING INFORMATION: Production records for an automobile manufacturer show the following figures for production per shift (maximum production is 720 cars per shift): 693 661 716 682 630 705 706 707 693 693 672 696 699 669 635 693 552 684 708 713 693 704 702 672 708 708 194. Would the mode be a useful summary statistic for these data? Why? 195. Find the median. 196. Find the mean. 197. What does the relation between the mean and median indicate about the shape of the data? 44 Describing Data: Numerical 198. Find the standard deviation. 199. How well does the Empirical Rule work for the fraction of data falling within one standard deviation of the mean? 200. Find the IQR. 201. Summarize the Empirical Rule. 202. For a particular sample of 50 scores on a statistics exam, the following results were obtained: Mean = 78 Mode = 84 First quartile = 68 Standard deviation = 11 Median = 80 Range = 52 Third quartile = 94 What score was earned by more students than any other score? Why? 203. For a particular sample, the mean is 3.7 and the standard deviation is 1.2. A new sample is formed by adding 6.3 to every item of data in the original sample. Find the mean and standard deviation of the new sample. QUESTIONS 204 THROUGH 209 ARE BASED ON THE FOLLOWING INFORMATION: Starting with a sample of two values 70 and 100, add three data values to your sample to obtain a new sample with certain statistics. 204. What are the three data values such that the new sample has a mean of 100? Justify your answer. 205. What are the three data values such that the new sample has a median of 70? Justify your answer. 206. What are the three data values such that the new sample has a mode of 87? Justify your answer. 45 Chapter 3 207. What are the three data values such that the new sample has a mean of 100 and a median of 70? Justify your answer. 208. What are the three data values such that the new sample has a mean of 100 and a mode of 87? Justify your answer. 209. What are the three data values such that the new sample has a mean of 100, a median of 70, and a mode of 87? Justify your answer. 210. For the following three samples, for which sample is the data most closely grouped about the sample mean? Give a written explanation that supports your conclusion. Sample 1: Sample 2: Sample 3: 211. 15, 16, 19, 21, 28; 44, 49, 50, 51, 57; and 122.8, 123.7, 124.6, 130.5, 135.8. Consider the following two sets of data: Set 1: 45 Set 2: 35 55 50 50 65 48 47 52 53 Compare the following measures for both sets: x , (x x ) , (x x ) 2 , and the range. Comment on the meaning of these comparisons. 212. The following subscripted x's represent a sample of size n = 67 which has been ranked from smallest ( x1 ) to largest ( x67 ) : x1 , x2 , x3 , x65 , x66 , x67 . . Prepare a 5number summary for this sample in terms of the subscripted x's. 213. A sample has a mean of 100.0 and a standard deviation of 15.0. According to Chebychev's Theorem, at least 8/9 of all of the data will lie between what two values? 214. A sample of size 50 has a mean of 60.0 and a standard deviation of 10.0. According to Chebychev's Theorem, at least what percent of the data is between 10 and 110? 46 Describing Data: Numerical 215. A sample of size 100 from a normal population has a mean of 110 and a standard deviation of 10.0. Using the Empirical Rule, about how many items of the sample will be above 130? 216. A large sample is selected from a normal distribution. The middle 99.7% of the sample data falls between 24.2 and 69.2. Estimate the sample mean and the sample standard deviation. QUESTIONS 217 THROUGH 219 ARE BASED ON THE FOLLOWING INFORMATION: Consider the following observations on shear strength of a joint bonded in a particular manner: 30.0 4.4 33.1 66.7 81.5 22.2 40.4 16.4 73.7 36.6 109.9 217. Determine the value of the sample mean. 218. Determine the value of the sample median. 219. Why is the median so different from the mean? QUESTIONS 220 THROUGH 223 ARE BASED ON THE FOLLOWING INFORMATION: A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (sec) to complete the escape: 373 424 370 325 364 394 366 402 364 392 325 369 339 374 393 359 356 356 359 403 363 334 375 397 220. Construct a stem-and-leaf display of the data. How does it suggest that the sample mean and median will compare? 221. Calculate the values of the sample mean and median. 222. By how much could the largest time, currently 424, be increased without affecting the value of the sample median? By how much could this value be decreased without affecting the value of the sample mean? 47 Chapter 3 223. What are the values of x and x when the observations are re-expressed in minutes? QUESTIONS 224 AND 225 ARE BASED ON THE FOLLOWING INFORMATION: Calculate the following sample observations on fracture strength: 128, 131, 142, 168, 87, 93, 105, 114, 96, and 98. 224. Calculate and interpret the value of the sample mean 225. Calculate and interpret the value of the sample standard deviation, 226. The first four deviations from the mean in a sample of n = 5 reaction times were .6, .9, 1.0, and 1.5. What is the fifth deviation from the mean? Give a sample for which these are the five deviations from the mean. QUESTIONS 227 THROUGH 229 ARE BASED ON THE FOLLOWING INFORMATION: A sample of eight doctors was asked how many flu shots they had given to patients this fall. The numbers of flu shots were 6, 3, 5, 24, 2, 6, 0, and 8. 227. Find the sample mean. 228. Find the median time to learn this task. 229. Based on the values of the mean and median in Questions (227) and (228) above, are the measurements symmetric or skewed? Why? QUESTIONS 230 THROUGH 232 ARE BASED ON THE FOLLOWING INFORMATION: The following data represent scores on a 15 point aptitude test: 8, 10, 15, 12, 14, and 13. 230. Subtract 5 from every observation and compute the sample mean for the original data and the new data. 231. Subtract 5 from every observation and complete the sample variance for the original data and the new data. 48 Describing Data: Numerical 232. What effect, if any, does subtracting 5 from every observation have on the mean and sample variance? sample QUESTIONS 233 THROUGH 236 ARE BASED ON THE FOLLOWING INFORMATION: A sample of 33 students were asked to rate themselves on whether they were outgoing or not using this five point scale: 1 = extremely extroverted, 2 = extroverted, 3 = neither extroverted nor introverted, 4 = introverted, or 5 = extremely introverted. The results are shown in the table below: Rating xi Frequency fi 1 2 3 4 5 1 7 20 5 0 233. Calculate the sample mean. 234. Calculate the median. 235. Calculate the sample standard deviation. 236. Find the percentage of measurements in the intervals x s and x 2s . Compare these results with the Empirical Rule percentages, and comment on the shape of the distribution. QUESTIONS 237 THROUGH 240 ARE BASED ON THE FOLLOWING INFORMATION: The following data represents the number of minutes an athlete spends training per day. 73 84 92 74 84 92 76 85 93 77 86 97 79 86 98 79 87 98 83 87 81 84 88 82 88 91 The mean and standard deviation were computed to be 85.54 and 6.97, respectively. 237. Create a stem and leaf plot for the distribution of training times. Is the distribution relatively mound-shaped? 238. What percentage of measurements would you expect to be between 71.60 and 99.48? 49 Chapter 3 239. What percentage of the measurements lies in the interval (71.60, 99.48)? 240. Do your answers to Questions 238 and 239 agree? If not, what conclusion can be drawn? QUESTIONS 241 AND 242 ARE BASED ON THE FOLLOWING INFORMATION: In a time study, conducted at a manufacturing plant, the length of time to complete a specified operation is measured for each on n = 40 workers. The mean and standard deviation are found to be 15.2 and 1.40, respectively. 241. Describe the sample data using the Empirical Rule. 242. Describe the sample data using Chebychev's Theorem. 50