Solutions Hmwk

ISDS-361AExercises 1. Mean, Median, Mode, Range, Variance & SD Panayiotis Skordi 1. A basketball player has the following points for seven games: 20, 25, 32, 18, 19, 22, 30 Compute the following measures of central location: a. mean b. median c. mode Answers a. Mean[pic] b. Median[pic] c. Mode There is no mode. 2. Consider the following population of measurements: 162, 152, 177, 157, 184, 176, 165, 181, 170, 163 a. compute the mean b. compute the median Answers a. Mean[pic] b. Median [pic] [pic] 3. The following data represent the number of children in a sample of 10 families from a certain community: , 2, 1, 1, 5, 3, 0, 1, 0, 2 a. compute the range b. compute the variance c. compute the standard deviation Answers a. Range [pic] b. Variance[pic] Remember that [pic] [pic][pic][pic][pic] 14 2. 14. 41 22 0. 10. 01 31-0. 90. 81 41-0. 90. 81 55 3. 19. 61 63 1. 11. 21 70-1. 93. 61 81-0. 90. 81 90-1. 93. 61 102 0. 10. 01 Total1924. 9 [pic] Method 2[pic] [pic][pic][pic] 1416 224 311 411 5525 639 700 811 900 1024 Total1961 [pic] This is the same answer – as we would expect. c. The standard deviation is [pic] 4. Consider the following population of measurements: 162, 152, 177, 157, 184, 176, 165, 181, 170, 163 a. ompute the standard deviation Answer [pic] Remember that [pic] [pic][pic][pic][pic] |1 |162 | |-6. 70 | |44. 89 | |2 |152 | |-16. 70 | |278. 89 | |3 |177 | |8. 30 | |68. 89 | |4 |157 | |-11. 70 | |136. 89 | |5 |184 | |15. 30 | |234. 09 | |6 |176 | |7. 30 | |53. 29 | |7 |165 | |-3. 0 | |13. 69 | |8 |181 | |12. 30 | |151. 29 | |9 |170 | |1. 30 | |1. 69 | |10 |163 | |-5. 70 | |32. 49 | | | | | | | | | | | | | | | |Total |1687. 0 | | | |1016. 10 | [pic] The standard deviation is the square root of the variance [pic] Method 2[pic] pic][pic][pic] |1 |162 | |26244 | |2 |152 | |23104 | |3 |177 | |31329 | |4 |157 | |24649 | |5 |184 | |33856 | |6 |176 | |30976 | |7 |165 | |27225 | |8 |181 | |32761 | |9 |170 | |28900 | |10 |163 | |26569 | | | | | | | | | | | | |1687 | |285613 | [pic] As before, the standard deviation is the square root of the variance pic] 5. Monthly rent data in dollars for a sample of one-bedroom apartments in a small town in Iowa are as follows: 220, 216, 220, 205, 210, 240, 195, 235, 204 and 205. a. compute the sample monthly average rent. b. compute the sample median. c. what is the mode? Answers a. Average [pic] [pic][pic] |1 |220 | |2 |216 | |3 |220 | |4 |205 | |5 |210 | |6 |240 | |7 |195 | |8 |235 | |9 |204 | |10 |205 | | | | | |2,150 | | | [pic] b. MEDIAN 195 204 205 205 210 216 220 220 235 240 [pic] c. MODE205 and 220 (bimodal) 6. A sample of 25 families was asked how many pets they owned. Their responses were summarized in the following table: |Number of Pets |0 |1 |2 |3 |4 |5 | |Number of Families |3 |10 |5 |4 |2 |1 | Determine the mean, median and the mode of the number of pets owned per family. ANSWERS a. MEAN[pic] [pic][pic][pic][pic] 0 |3 |0 | |1 |10 |10 | |2 |5 |10 | |3 |4 |12 | |4 |2 |8 | |5 |1 |5 | | | | | | |25 |45 | [pic] b. MEDIAN1 median is 13th observation. c. MODE 1 most families have 1 pet. 7.The following data represent the ages in years of a sample of 25 employees from a government department: 31,43,56,23,49,42,33,61,44,28,48,38,44,35,40,64,52,42,47,39,53,27,36,35 and 20. Construct a relative frequency distribution for the data, using five class intervals and the value 20 as the lower limit of the first class. |20 | | | |23 | | | |27 | | | |28 | |4 | |31 | | | |33 | | | |35 | | | 35 | | | |36 | | | |38 | | | |39 | |7 | |40 | | | |42 | | | |42 | | | |43 | | | |44 | | | |44 | | | |47 | | | |48 | | | |49 | |9 | |52 | | | 53 | | | |56 | |3 | |61 | | | |64 | |2 | |Class Limits |Frequency |Relative Frequency | | | | | |20 upto 30 |4 |0. 16 | |30 upto 40 |7 |0. 28 | |40 upto 50 |9 |0. 36 | |50 upto 60 |3 |0. 12 | |60 upto 70 |2 |0. 8 | | | | | | |25 |1. 00 | 8. How many hours a day do college students spend studying? That question was asked to 100 college students, and the data are p resented in the table below. Hours per DayFrequency 0-225 2-448 4-615 6-89 8-103 a. compute the range b. compute the average hours per day spent studying. c. compute the variance of the number of hours per day spent studying d. find the median e. find the mode Answers a. The range is 10-0 = 10 b. Average[pic] [pic][pic][pic][pic][pic] 1 |0-2 |25 |1 |25 | |2 |2-4 |48 |3 |144 | |3 |4-6 |15 |5 |75 | |4 |6-8 |9 |7 |63 | |5 |8-10 |3 |9 |27 | | | | | | | | | |100 | |334 | [pic] c. compute the variance of the number of hours per day spent studying [pic] note that [pic] [pic][pic][pic][pic][pic][pic][pic] |1 |0-2 |25 |1 |-2. 34 |5. 4756 |136. 89 | |2 |2-4 |48 |3 |-0. 34 |0. 1156 |5. 488 | |3 |4-6 |15 |5 |1. 66 |2. 7556 |41. 334 | |4 |6-8 |9 |7 |3. 66 |13. 3956 |120. 5604 | |5 |8-10 |3 |9 |5. 66 |32. 0356 |96. 1068 | | | | | | | | | | | |100 | | |53. 7780 |400. 4400 | [pic] Method 2 [pic] [pic][pic][pic][pic][pic][pic] 1 |0-2 |25 |1 |25 |25 | |2 |2-4 |48 |3 |144 |432 | |3 |4-6 |15 |5 |75 |37 5 | |4 |6-8 |9 |7 |63 |441 | |5 |8-10 |3 |9 |27 |243 | | | | | | | | | | |100 | |334 |1516 | [pic] d. Find the MEDIAN3 e. Find the MODE3 9. The following data is for the daily room rate in $, for staying at various hotels in downtown LA Hotels Offering RateDaily room rate 130 50 1470 690 3110 4130 1150 1170a. Compute the range b. Compute the average daily room rate. c. Compute the variance of the room rate. d. Find the median. e. Find the mode. Answers a. Range 170-30 =140 b. Average[pic]where [pic] [pic][pic][pic][pic] | 1 |1 |30 |30 | |2 |7 |50 |350 | |3 |14 |70 |980 | |4 |6 |90 |540 | |5 |3 |110 |330 | |6 |4 |130 |520 | |7 |1 |150 |150 | 8 |1 |170 |170 | | | | | | | |37 | |3070 | [pic] c. Variance of the room rate [pic] [pic][pic][pic][pic][pic][pic] |1 |1 |30 |-52. 9730 |2806. 1359 |2806. 1359 | |2 |7 |50 |-32. 9730 |1087. 2169 |7610. 5186 | |3 |14 |70 |-12. 9730 |168. 2980 |2356. 1724 | |4 |6 |90 |7. 0270 |49. 3791 |296. 747 | |5 |3 |110 |27. 0270 |730. 4602 |2191. 3806 | |6 |4 |130 |47. 0270 |2211. 5413 |8846. 1651 | |7 |1 |150 |67. 0270 |4492. 6224 |4492. 6224 | |8 |1 |170 |87. 0270 |7573. 7034 |7573. 7034 | | | | | | | | | |37 | |136. 2162 | |36172. 9730 | [pic]Method 2 [pic] [pic][pic][pic][pic][pic] |1 |1 |30 |30 |900 | |2 |7 |50 |350 |17500 | |3 |14 |70 |980 |68600 | |4 |6 |90 |540 |48600 | |5 |3 |110 |330 |36300 | |6 |4 |130 |520 |67600 | |7 |1 |150 |150 |22500 | |8 |1 |170 |170 |28900 | | | | | | | | |37 | |3,070 |290,900 | [pic] d. Find the MEDIAN70 e. Find the mode70 Solutions Hmwk ISDS-361AExercises 1. Mean, Median, Mode, Range, Variance & SD Panayiotis Skordi 1. A basketball player has the following points for seven games: 20, 25, 32, 18, 19, 22, 30 Compute the following measures of central location: a. mean b. median c. mode Answers a. Mean[pic] b. Median[pic] c. Mode There is no mode. 2. Consider the following population of measurements: 162, 152, 177, 157, 184, 176, 165, 181, 170, 163 a. compute the mean b. compute the median Answers a. Mean[pic] b. Median [pic] [pic] 3. The following data represent the number of children in a sample of 10 families from a certain community: , 2, 1, 1, 5, 3, 0, 1, 0, 2 a. compute the range b. compute the variance c. compute the standard deviation Answers a. Range [pic] b. Variance[pic] Remember that [pic] [pic][pic][pic][pic] 14 2. 14. 41 22 0. 10. 01 31-0. 90. 81 41-0. 90. 81 55 3. 19. 61 63 1. 11. 21 70-1. 93. 61 81-0. 90. 81 90-1. 93. 61 102 0. 10. 01 Total1924. 9 [pic] Method 2[pic] [pic][pic][pic] 1416 224 311 411 5525 639 700 811 900 1024 Total1961 [pic] This is the same answer – as we would expect. c. The standard deviation is [pic] 4. Consider the following population of measurements: 162, 152, 177, 157, 184, 176, 165, 181, 170, 163 a. ompute the standard deviation Answer [pic] Remember that [pic] [pic][pic][pic][pic] |1 |162 | |-6. 70 | |44. 89 | |2 |152 | |-16. 70 | |278. 89 | |3 |177 | |8. 30 | |68. 89 | |4 |157 | |-11. 70 | |136. 89 | |5 |184 | |15. 30 | |234. 09 | |6 |176 | |7. 30 | |53. 29 | |7 |165 | |-3. 0 | |13. 69 | |8 |181 | |12. 30 | |151. 29 | |9 |170 | |1. 30 | |1. 69 | |10 |163 | |-5. 70 | |32. 49 | | | | | | | | | | | | | | | |Total |1687. 0 | | | |1016. 10 | [pic] The standard deviation is the square root of the variance [pic] Method 2[pic] pic][pic][pic] |1 |162 | |26244 | |2 |152 | |23104 | |3 |177 | |31329 | |4 |157 | |24649 | |5 |184 | |33856 | |6 |176 | |30976 | |7 |165 | |27225 | |8 |181 | |32761 | |9 |170 | |28900 | |10 |163 | |26569 | | | | | | | | | | | | |1687 | |285613 | [pic] As before, the standard deviation is the square root of the variance pic] 5. Monthly rent data in dollars for a sample of one-bedroom apartments in a small town in Iowa are as follows: 220, 216, 220, 205, 210, 240, 195, 235, 204 and 205. a. compute the sample monthly average rent. b. compute the sample median. c. what is the mode? Answers a. Average [pic] [pic][pic] |1 |220 | |2 |216 | |3 |220 | |4 |205 | |5 |210 | |6 |240 | |7 |195 | |8 |235 | |9 |204 | |10 |205 | | | | | |2,150 | | | [pic] b. MEDIAN 195 204 205 205 210 216 220 220 235 240 [pic] c. MODE205 and 220 (bimodal) 6. A sample of 25 families was asked how many pets they owned. Their responses were summarized in the following table: |Number of Pets |0 |1 |2 |3 |4 |5 | |Number of Families |3 |10 |5 |4 |2 |1 | Determine the mean, median and the mode of the number of pets owned per family. ANSWERS a. MEAN[pic] [pic][pic][pic][pic] 0 |3 |0 | |1 |10 |10 | |2 |5 |10 | |3 |4 |12 | |4 |2 |8 | |5 |1 |5 | | | | | | |25 |45 | [pic] b. MEDIAN1 median is 13th observation. c. MODE 1 most families have 1 pet. 7.The following data represent the ages in years of a sample of 25 employees from a government department: 31,43,56,23,49,42,33,61,44,28,48,38,44,35,40,64,52,42,47,39,53,27,36,35 and 20. Construct a relative frequency distribution for the data, using five class intervals and the value 20 as the lower limit of the first class. |20 | | | |23 | | | |27 | | | |28 | |4 | |31 | | | |33 | | | |35 | | | 35 | | | |36 | | | |38 | | | |39 | |7 | |40 | | | |42 | | | |42 | | | |43 | | | |44 | | | |44 | | | |47 | | | |48 | | | |49 | |9 | |52 | | | 53 | | | |56 | |3 | |61 | | | |64 | |2 | |Class Limits |Frequency |Relative Frequency | | | | | |20 upto 30 |4 |0. 16 | |30 upto 40 |7 |0. 28 | |40 upto 50 |9 |0. 36 | |50 upto 60 |3 |0. 12 | |60 upto 70 |2 |0. 8 | | | | | | |25 |1. 00 | 8. How many hours a day do college students spend studying? That question was asked to 100 college students, and the data are p resented in the table below. Hours per DayFrequency 0-225 2-448 4-615 6-89 8-103 a. compute the range b. compute the average hours per day spent studying. c. compute the variance of the number of hours per day spent studying d. find the median e. find the mode Answers a. The range is 10-0 = 10 b. Average[pic] [pic][pic][pic][pic][pic] 1 |0-2 |25 |1 |25 | |2 |2-4 |48 |3 |144 | |3 |4-6 |15 |5 |75 | |4 |6-8 |9 |7 |63 | |5 |8-10 |3 |9 |27 | | | | | | | | | |100 | |334 | [pic] c. compute the variance of the number of hours per day spent studying [pic] note that [pic] [pic][pic][pic][pic][pic][pic][pic] |1 |0-2 |25 |1 |-2. 34 |5. 4756 |136. 89 | |2 |2-4 |48 |3 |-0. 34 |0. 1156 |5. 488 | |3 |4-6 |15 |5 |1. 66 |2. 7556 |41. 334 | |4 |6-8 |9 |7 |3. 66 |13. 3956 |120. 5604 | |5 |8-10 |3 |9 |5. 66 |32. 0356 |96. 1068 | | | | | | | | | | | |100 | | |53. 7780 |400. 4400 | [pic] Method 2 [pic] [pic][pic][pic][pic][pic][pic] 1 |0-2 |25 |1 |25 |25 | |2 |2-4 |48 |3 |144 |432 | |3 |4-6 |15 |5 |75 |37 5 | |4 |6-8 |9 |7 |63 |441 | |5 |8-10 |3 |9 |27 |243 | | | | | | | | | | |100 | |334 |1516 | [pic] d. Find the MEDIAN3 e. Find the MODE3 9. The following data is for the daily room rate in $, for staying at various hotels in downtown LA Hotels Offering RateDaily room rate 130 50 1470 690 3110 4130 1150 1170a. Compute the range b. Compute the average daily room rate. c. Compute the variance of the room rate. d. Find the median. e. Find the mode. Answers a. Range 170-30 =140 b. Average[pic]where [pic] [pic][pic][pic][pic] | 1 |1 |30 |30 | |2 |7 |50 |350 | |3 |14 |70 |980 | |4 |6 |90 |540 | |5 |3 |110 |330 | |6 |4 |130 |520 | |7 |1 |150 |150 | 8 |1 |170 |170 | | | | | | | |37 | |3070 | [pic] c. Variance of the room rate [pic] [pic][pic][pic][pic][pic][pic] |1 |1 |30 |-52. 9730 |2806. 1359 |2806. 1359 | |2 |7 |50 |-32. 9730 |1087. 2169 |7610. 5186 | |3 |14 |70 |-12. 9730 |168. 2980 |2356. 1724 | |4 |6 |90 |7. 0270 |49. 3791 |296. 747 | |5 |3 |110 |27. 0270 |730. 4602 |2191. 3806 | |6 |4 |130 |47. 0270 |2211. 5413 |8846. 1651 | |7 |1 |150 |67. 0270 |4492. 6224 |4492. 6224 | |8 |1 |170 |87. 0270 |7573. 7034 |7573. 7034 | | | | | | | | | |37 | |136. 2162 | |36172. 9730 | [pic]Method 2 [pic] [pic][pic][pic][pic][pic] |1 |1 |30 |30 |900 | |2 |7 |50 |350 |17500 | |3 |14 |70 |980 |68600 | |4 |6 |90 |540 |48600 | |5 |3 |110 |330 |36300 | |6 |4 |130 |520 |67600 | |7 |1 |150 |150 |22500 | |8 |1 |170 |170 |28900 | | | | | | | | |37 | |3,070 |290,900 | [pic] d. Find the MEDIAN70 e. Find the mode70