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M_BANK\YR11-GEN\DATA04.HSC Summary Statistics 1)! MIS84-A6i A factory which manufactures shoes sells direct to the public and records the sales of different sizes over a period of one week. The results of this survey are shown in the table below. Shoe Size Number Sold a. b. c d. 6 8 6·5 8 7 8 7·5 11 8 14 8·5 7 9 6 9·5 5 10 2 From this data, calculate the mean show size and the modal shoe size. On the graph paper provided, draw a cumulative frequency polygon to represent the data. Label your axes clearly. Using the graph from part (b), or otherwise, find the median shoe size. The manager of the factory wishes to know which size is sold most often. Which measure - the mean, the median or the mode - will be of most value to him? Give a reason for your answer.¤ 70 60 50 CUMULATIVE 40 FREQUENCY 30 20 10 5 6 7 8 9 10 SHOE SIZE « a) 7·70 (to 2 d.p.) b) 2)! c) 7·5 d) The mode. This gives the shoe size sold most often. » MIS85-A6 The tables below show the scores compiled by three teams in ten innings in a series of oneday limited-over cricket matches. Team A 206 212 287 209 205 257 205 261 277 290 i. ii. Team B 212 273 263 280 257 222 234 233 208 227 Team C 163 191 201 204 197 210 360 201 195 189 Calculate the range, mean and standard deviation for each set of team scores. Which one of the three teams had the most consistent scores? Give a reason for your answer. ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 96 M_BANK\YR11-GEN\DATA04.HSC iii. Which one of the above thirty scores was least representative of a team's actual performance? Give a reason for your answer. iv. For team C, calculate the median score. v. For team C, which measure - the mean or the median - best describes the team's performance? Give a reason for your answer.¤ « i) TEAM A TEAM B TEAM C RANGE 85 72 197 MEAN 240·9 240·9 211·1 STANDARD DEVIATION 34·9 24·3 51·1 ii) Team B. Its range and standard deviation are smaller iii) Team C’s score of 360. It is almost 3 standard deviations above the mean. iv) 199 v) The median. The mean is affected by the score of 360 » 3)! MIS86-B6ii A light bulb manufacturer tests light bulbs by determining the number of hours for which bulbs will burn continuously. The results of testing 30 bulbs is presented in the histogram below. 10 9 8 7 Number of 6 bulbs 5 4 3 2 1 a. 8 9 10 11 12 13 14 15 Life of bulb in hundreds of hours Draw up a frequency table to represent this information using the headings shown in the diagram below. Life of bulbs b. Number of bulbs For this distribution calculate . The mean life of a bulb. . The mode. . The range. ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 97 M_BANK\YR11-GEN\DATA04.HSC . . The median. The standard deviation.¤ « a) Life of Bulbs Number of Bulbs 800 1 900 2 1000 4 1100 9 1200 6 1300 5 1400 2 1500 1 b) ) 1150 hours ) 1100 hours ) 700 hours ) 1100 hours ) 156·5 hours » 4)! MIS89-A6b The heights in centimetres of a sample of 25 girls in New South Wales are recorded in 7 classes in the table below. HEIGHT RANGE (cm) 140 - 149 150 - 159 160 - 169 170 - 179 180 - 189 190 - 199 200 - 209 MID POINT (cm) 144·5 154·5 164·5 174·5 184·5 194·5 204·5 FREQUENCY 1 11 9 ........... 1 0 1 i. The frequency for the 170 - 179 cm class is missing. How do we know it must be 2? ii. What is the modal class? iii. Calculate the mean. iv. Calculate the standard deviation.¤ « i) We are told there are 25 girls in the sample ie. the total frequency must be 25 ii) 150 - 159 cm iii) 162·5 cm iv) 12 cm » 5)! MIS90-A6c Motorists ring a Road Service Patrol for assistance when their car breaks down. The number of breakdown calls received by the Road Service Patrol was recorded over a period of 40 days. The number of breakdown calls is shown below: 70 54 65 48 i. 50 48 48 63 55 55 70 50 59 63 69 58 50 70 63 65 62 59 62 59 70 60 67 66 71 60 69 62 54 60 50 60 59 70 55 70 Copy the table below into your examination booklet and complete it. ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 98 M_BANK\YR11-GEN\DATA04.HSC Class Interval 47-51 52-56 57-61 62-66 67-71 ii. iii. Class Centre Tally Write down the modal class. Calculate the mean number of breakdown calls received each day.¤ Class Interval 47-51 52-56 57-61 62-66 67-71 6)! 7)! 0 12 1 9 What was the average number of errors per page? (A) 1·6 (B) 2 (C) 8 9)! Class Centre 49 54 59 64 69 Tally Freq. « i) fx 7 343 5 270 9 531 9 576 10 690 TOTAL 40 2410 ii) 67 - 71 iii) 60·25 » MIS93-A9 A spelling test was given to a group of Year Six students and to a group of Year Nine students. The 30 Year Six students had an average score of 13, and the 60 Year Nine students had an average score of 19. What was the average score of the combined groups? (A) 16 (B) 17 (C) 32 (D) 45¤ « B » MIS93-A20 The number of errors on each page of a document was recorded. The results are shown in the table below. Number of errors Number of pages 8)! Frequency 2 7 3 7 4 5 (D) 12·8¤ « A » MIS95-A15 Three students have an average mass of 45 kg. A fourth student with a mass of 66 kg joins the group. What is the average mass of the four students? (A) 48 kg (B) 51 kg (C) 56 kg (D) 62 kg¤ « B » MIS96-A4 Which one of the following groups of scores has a mean of 60 and a median of 50? (A) 10, 50, 60, 70, 80, 90 (B) 40, 40, 45, 55, 70, 90 (C) 40, 45, 45, 55, 85, 90 (D) 30, 40, 50, 50, 70, 80¤ « C » ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 99 M_BANK\YR11-GEN\DATA04.HSC 10)! 11)! MIS96-A14 Three Mathematics classes did the same assessment task. The mean marks for the three classes were 60, 75, and 76·5. The numbers of students in the three classes were 22, 18, and 20 respectively. What was the mean mark for all students on this assessment task? (A) 45·25 (B) 70 (C) 70·5 (D) 75¤ « B » MIS97-A8 Score Frequency 21 2 22 4 23 6 24 1 25 1 A score of 25 is added to this sample. Which of these measures will change? (A) Range (B) Median (C) Mode (D) Mean¤ 12)! 13)! 14)! 15)! « D » MIS97-A16 After five English tests, Sue’s mean mark was 65. In the next three English tests she scored 70, 75 and 80. Calculate Sue’s mean mark for all of these English tests. (A) 68·75 (B) 70 (C) 72·5 (D) 75¤ « A » MIS98-2 In which set of scores is the mode greater than the median? (A) 3, 4, 4, 5, 5, 6, 6, 6, 7 (B) 3, 4, 4, 5, 5, 5, 6, 6, 7 (C) 3, 4, 4, 4, 5, 5, 6, 6, 7 (D) 2, 2, 2, 2, 3, 4, 4, 5, 6¤ « A » MIS98-14 The mean height of five Sydney Flames basketball players at the start of a game is 1·88 m. During the game, a player who is 1·74 m tall is injured and is replaced by a player who is 1·94 m tall. What is the mean height of the 5 players now? (A) 1·89 m (B) 1·91 m (C) 1·92 m (D) 2·08 m¤ « C » MIS98-23b The Cavetto Pasta Company employs seventy people. The annual income of the employees is shown in the frequency distribution table below. Annual income ($) 10 000 - 19 999 20 000 - 29 999 30 000 - 39 999 40 000 - 49 999 50 000 - 59 999 60 000 - 69 999 Class centre x 15 000 25 000 35 000 45 000 55 000 65 000 Number of employees F 16 24 11 9 7 3 f = 70 ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 100 fx 240 000 600 000 A 405 000 385 000 195 000 fx = M_BANK\YR11-GEN\DATA04.HSC i. ii. iii. iv. v. Calculate the value of A in the fx column. Calculate fx. Determine the mean annual income. In which annual income class does the median of this distribution lie? Using the information in the table and your calculator, find the standard deviation of this distribution. vi. Cavetto Pasta wishes to employ another eight people. Four of these new employees will earn $15 000 p.a. each, and the other four will earn $65 000 p.a. each. What will be the effect of these new employees on the standard deviation of income distribution at Cavetto Pasta? Give a brief reason for your answer.¤ « i) 385 000 ii) 2 210 000 iii) $31 571 iv) 20 000 - 29 999 v) 14 331 (to nearest whole number) vi) The standard deviation will increase » 16)! MIS99-12 After six Mathematics tests, Nadine’s mean score was 71. She hopes to raise her mean score to 75 after the next test. What mark will Nadine need to achieve on her seventh test if her mean mark is to be 75? (A) 75 (B) 79 (C) 95 (D) 99¤ « D » 17)! MIS99-16 50 No. of 40 students 30 20 10 18)! 0 1 2 3 4 5 Quiz mark The graph shows the results of 150 students on a Maths quiz. Which statement is correct? (A) The median is 3 and the mode is 2 (B) The median is 2 and the mode is 2 (C) The median is 2 and the mode is 3 (D) The median is 3 and the mode is 3¤ « B » MIS99-23a The maximum temperature for each day of September was recorded. The information is listed in the table below. Temperature (C) x 19 20 21 22 23 24 i. Frequency f 4 6 9 0 7 4 fx 76 120 A 0 161 96 What is the mode of this distribution? ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 101 Cumulative Frequency 4 10 19 19 B 30 M_BANK\YR11-GEN\DATA04.HSC ii. iii. iv. v. vi. Find the value of A in the fx column. Calculate the mean maximum temperature for the month of September. Find the standard deviation of this distribution. Find the value of B in the Cumulative Frequency column. Complete the cumulative frequency histogram below. 35 30 25 20 Cumulative frequency 15 10 5 0 vii. viii. 19 20 21 22 23 Temperature (C) Draw a cumulative frequency polygon on the same graph. Use the graph to find the interquartile range of this distribution.¤ ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 102 24 M_BANK\YR11-GEN\DATA04.HSC « i) 21 ii) 189 iii) 21·4C iv) 1·65 v) 26 vi) vii) 35 30 25 Cumulative frequency 20 15 10 5 0 19)! 20)! 19 23 22 21 Temperature (C) 20 24 viii) 2·9 » MIS00-15 In order to pass his university course, Dimitri must average at least 50% over 5 assessment tasks. After the first 4 assessment tasks, Dimitri has a mean mark of 45%. All tasks have equal weight. What is the minimum mark, out of 100, that Dimitri must score in the fifth assessment task to pass the course? (A) 50 (B) 55 (C) 65 (D) 70¤ « D » MIS00-23c The following table contains data about the height of 40 year 12 students. Height range (cm) 155-159 Class centre (cm) 157 ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 103 Frequency 3 M_BANK\YR11-GEN\DATA04.HSC 160-164 165-169 170-174 175-179 180-184 162 167 172 177 182 185-189 190-194 195-199 187 192 197 7 6 6 9 A 3 2 1 The frequency for the 180-184 height range is written as A in the table. What is the value of A ? ii. What is the modal class? iii. What is the median class? iv. Use the class centres to calculate the approximate mean height. ¤ « i) 3 ii) 175 - 179 iii) 170 - 174 iv) 173 cm » GEN01-8 The following frequency table shows Ravdeep’s scores on a number of quizzes. i. 21)! Score 1 2 3 4 5 Frequency 2 3 5 2 1 Which expression gives Ravdeep’s mean score? 2 6 15 8 5 2 6 15 8 5 (A) (B) 13 5 1 2 3 4 5 1 2 3 4 5 (C) (D) ¤ 13 5 22)! 23)! «A » GEN02-26a After three small quizzes, Vicki has an average mark of 5. She wants to increase her average to 6. What mark must she score in the next quiz for her average mark to be exactly 6? ¤ « 9 » GEN02-26b Roxy selected 30 students at random from Year 12 at her high school, and asked each of them how many text messages they had sent from a mobile phone within the last day. The results are summarised in the following table. Number of text messages sent 0 1 Frequency 3 3 ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 104 M_BANK\YR11-GEN\DATA04.HSC 2 3 4 5 4 4 9 7 i. 24)! Calculate the mean number of text messages sent. (Give your answer correct to two decimal places.) ii. Calculate the sample standard deviation. (Give your answer correct to two decimal places.) iii. Determine the median number of text messages sent. iv. Describe the skewness of the data. v. There are 150 students in Year 12 at Roxy’s school. Use the sample data in the table to estimate how many of these Year 12 students would have sent more than three text messages within the last day. ¤ « i) 313 ii) 166 iii) 4 iv) The data are negatively skewed v) 80 » GEN03-12 Joy asked the Students in her class how many brothers they had. The answers were recorded in a frequency table as follows: Number of brothers 0 1 2 3 4 Frequency 5 10 3 1 1 What is the mean number of brothers? (A) 115 (B) 2 (C) 23 25)! (D) 4¤ «A » GEN03-25a A census was conducted of the 33 171 households in Sunnytown. Each household was asked to indicate the number of cars registered to that household. The results are summarised in the following table. Number of cars 0 1 2 3 4 Total i. ii. Frequency 2735 12 305 13 918 3980 233 33 171 1. Determine the mode number of cars in a household. 2. Explain what is meant by “the mode number of cars in a household”. Sunnytown Council issued a ‘free parking’ sticker for each car registered to ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 105 M_BANK\YR11-GEN\DATA04.HSC iii. iv. a household in Sunnytown. How many parking stickers were issued? The council represented the results of the census in a sector graph. What is the angle in the sector representing the households with no cars? Give your answer to the nearest degree. Visitors to Sunnytown Airport have to pay for parking. The following step graph shows the cost of parking for t hours. 30 25 Parking Fee ($) 20 15 10 5 26)! 27)! 28)! 0 1 3 6 12 What is the cost for a car that is parked one evening from 6 pm to 8:30 pm? ¤ « i) 1) 2 2) The most common number of cars per household ii) 53 013 iii) 30 iv) $10 » GEN04-7 What are the median and the mode of the set scores 1, 3, 3, 3, 4, 5, 7, 7, 12? (A) Median 3, mode 5 (B) Median 3, mode 3 (C) Median 4, mode 5 (D) Median 4, mode 3¤ « D » Gen05-1 What is the mean of the set of scores? 3, 4, 5, 6, 6, 8, 8, 8, 15 (A) 6 (B) 7 (C) 8 (D) 9¤ « B » Gen05-27d Nine students were selected at random from a school, and their ages were recorded. 12 Ages 11 16 14 16 15 14 15 14 i. What is the sample standard deviation, correct to two decimal places? ii. Briefly explain what is meant by the term standard deviation.¤ « i) 1∙69 ii) Standard deviation measures the average deviation of the scores from the mean » 29)! Gen06-4 A set of scores is displayed in a stem-and-leaf plot. ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 106 M_BANK\YR11-GEN\DATA04.HSC 1 2 3 4 2 5 8 1 2 8 9 3 What is the median of this set of scores? (A) 28 (B) 30 (C) 33 3 9 (D) 47¤ « C » 30)! 31)! Gen06-12 The mean of a set of 5 scores is 62. What is the new mean of the set of scores after a score of 14 is added? (A) 38 (B) 54 (C) 62 (D) 76¤ « B » Gen06-23c Vicki wants to investigate the number of hours spent on homework by students at her high school. i. Briefly describe a valid method of randomly selecting 200 students for a sample. ii. Vicki chooses her sample and asks each student how many hours (to the nearest hour) they usually spend on homework during one week. The responses are shown in the frequency table. Number of hours spent on homework in a week 0 to 4 5 to 9 10 to 14 15 to 19 Frequency 69 72 38 21 What is the mean amount of time spent on homework?¤ « i) Assign each student a umber and draw numbers from a hat ii) 7∙275 hours » ¤©BOARD OF STUDIES NSW 1984 - 2006 ©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006 107