# THOMAS WHITHAM SIXTH FORM

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1 THOMAS WHITHAM SIXTH FORM Handling Data Levels 6 8 S. J. Cooper Probability Tree diagrams & Sample spaces Statistical Graphs Scatter diagrams Mean, Mode & Median Year 9 B U R N L E Y C A M P U S, B U R N L E Y L A N C A S H I R E B B 1 0 1JD T EL:

2 Handling Data(1) 1. An ordinary pack of playing cards is shuffled and the top card selected. What is the probability that the top card is a) a 3 b) a picture card c) the 4 of clubs 2. (a) If twenty drawing pins are thrown in the air and 4 land point down, what is the probability of a drawing pin thrown landing point down? (b) If fifty drawing pins are thrown in the air and 12 land point down, what is the probability of a drawing pin thrown landing point down? (c) Which probability is more reliable in parts (a) and (b), give a reason for your answer. 3. A spinner has ten sides numbered 1 to 10. If the spinner is spun once what is the probability that the spinner lands on a) the number 9 b) a prime number c) a factor of A bag contains 50 balls numbered from 1 to 50. A ball is selected at random, what is the probability that the ball a) is numbered 40 b) is odd c) is a multiple of 6 5. A box contains three red balls, five green balls, two white balls and four blue balls. One ball is selected at random what is the probability that the ball is a) blue b) yellow? S J Cooper 1

3 6. Alex and Alice move into their new home, and as a practical joke their friends remove all the labels of their tinned food. If the cupboard contained thirty tins of which three tins were baked beans, four tins were vegetable soup, two tins were sliced carrots and the rest were dog food. What is the probability that Alex and Alice open a can of dog food for tea! 7. At the Summer fair 1000 raffle tickets were sold, if Mrs Prestley bought 20 tickets what is the probability a) of her winning first prize b) of her winning second prize given that someone else won first prize? 8. Adam throws a coin thought to be biased 10 times and obtained 4 heads, Brian throws the same coin 20 times and obtained 8 heads and Carl throws the coin 50 times and obtained 27 heads. Based on this information is the coin biased? Give a reason for your answer. 9. Balal thought that he could predict the number of red discs in a bag containing 20 discs. At first he selected a disc 10 times, replacing the disc each time, and recorded the number of times he selected a red disc. Next he repeated the experiment 20 times and the 40 times. His results are shown in the table below. Number of picks Number of red discs a) work out the relative probability for each experiment b) Based on this information how many red discs do you expect in the bag? Give a reason for your answer. S J Cooper 2

4 Handling Data(2) 1. A game consists throwing two dice numbered from 1 to 6. The score obtained is the found by subtracting the smallest number from the highest. a) What is the smallest score a player can obtain? b) What is the highest score a player can obtain? c) Copy and complete the table below First die Second die d) What is the probability that a player will obtain a score of (i) 3 (ii) 4 (iii) 5? 2. When selecting options for GCSE subjects in year 10, students are expected two selected two subjects from the list below. French Geography German ICT Drama PE History RE Sociology Business Studies Art Draw up a table of all possible combinations S J Cooper 3

5 3. A game consists of two spinners one divided into four equal points labelled 2, 4, 6, 8 and the other divided into five equal parts labelled 1, 3, 5, 7, 9. Both spinners are spun and the total score is found by adding the two scores together. a) Copy and complete the table below b) What is the probability of obtaining a score of (i) 6 (ii) a factor of 16 (iii) an even number (iv) a 12? 4. A fair coin and a fair die are thrown together. Obtain the probability that the coin lands on a head and the die on a six. 5. List the possible outcomes when a fair coin is thrown three times. a) what is the probability of obtaining three heads b) what is the probability of obtaining one head and two tails c) what is the probability of obtaining at least one head? S J Cooper 4

6 Handling Data(3) 1. Given that each week the probability that Sharon Osbourne will quit X factor is 0.3 copy and complete the tree diagram which shows all possible outcomes for the next two weeks 1 st week 2 nd week Quits Quits... Stays. What is the probability that Sharon will quit on two consecutive weeks? 2. A test consists of multiple choice questions with five choice answers of which only one is correct. In 40 questions Adam is unsure of only 2. If he simply randomly selects an answer for these two questions a) Copy and complete the tree diagram for his possible outcome 1 st question 2 nd question Correct Correct... Wrong. b) What is the probability that Adam answers these two questions correctly? S J Cooper 5

7 3. A bag contains 5 red balls and 3 green balls. A ball is drawn at random and then replaced. Another ball is drawn. a) Copy and complete the tree diagram below 1 st ball 2 nd ball Red Red... Green. b) What is the probability that both balls are green? 4. A bag contains 10 counters of which 3 are black and the rest are red. A counter is selected at random, replaced and a second counter is selected. a) Draw a tree diagram showing all possible outcomes. b) What is the probability of selected two counters of the same colour? 5. A committee is made up of 10 students of which 4 are boys. If two students are selected at random to represent chairperson and vice chairperson of the committee, what is the probability that both will be girls? 6. A bag contains 8 balls of which two are green and the rest are blue. If two balls are selected at random one after the other a) draw a tree diagram to show all possible outcomes b) what is the probability of selecting one of each colour? 7. A bag contains nine sweets of which four are mints and five are chocolate. Two are selected at random. What is the probability of selecting one mint and one chocolate? 8. A coin is thrown three times. What is the probability of obtaining three heads? S J Cooper 6

8 Handling Data (4) Mean 1. Find the mean for each of the following set of numbers a) 3, 7, 14, 21, 25 b) 16, 18, 21, 34, 38, 41 c) 67, 54, 37, 49, 55, 63, 67 d) 9, 75, 1, 16, 49, 84, 64, 88, 27, 14, 2 e) 1.3, 1.8, 1.7, 2.0, 1.8, 3.4 f) 6, 6, 7, 7, 7, 8, 8, 9, 9, 12, 12, 14, 14, 15, 19 g) 4.6, 2.7, 1, 8.8, 2.7, 3.5, 8.2, 4.6, 4.8, In monitoring the traffic coming into Burnley a police patrol car noted down the speed of the last 8 cars as shown below. 31, 27, 34, 48, 32, 37, 24, 31 Workout the mean speed for these eight cars. 3. In a recent maths test the teacher wanted to obtain the average mark for his class. The scores are given below. Find the class average. 87, 84, 82, 75, 74, 61, 74, 81, 54, 17, 34, 45, 77, 54, 38, Ten students are given a tube of smarties and asked to find the total number in the tube, the results are given below. Based on this evidence what is the average number of smarties per tube? 38, 37, 45, 42, 49, 39, 41, 41, The table below shows the number of exercise books each of twenty students brought to school on one Monday morning. Work out the mean number of books brought to school on Monday morning. Number of books Number of students S J Cooper 7

9 Handling data (5) calculating the mean 6. The table below shows the number of minutes each of 30 students were late to Tutor on one morning. Number of Minutes Number of students Calculate the mean length of time this group of students were late. 7. A ordinary six-sided die is thrown twenty times and the scores obtained are shown in the table below. Score frequency Calculate the mean score on the die. 8. In a paragraph of 50 words the word length of each word was recorded in the table below. Word length frequency Calculate the mean word length of this text. 9. On my way to work I have to pass 4 sets of traffic lights. The table below shows the number of times I was stopped on each of 40 days. No. of times stopped Frequency Calculate the mean number of lights I am stopped at on my way to work. 10. Alan is given homework every day at college and records the number of pieces he was given each day over the last 30 days. No. of Homeworks Frequency Calculate the mean number of pieces of homework Alan can expect on a given day. S J Cooper 8

10 Handling data (6) calculating the mean 11. The table below shows the time taken to complete the crossword puzzle by 20 teachers. Time taken Frequency Calculate an estimate for the mean length of time taken to complete the crossword puzzle. 12. The speeds of forty cars passing the school gates are recorded in the table below. Speed Frequency Calculate an estimate for the mean speed of the forty cars as they pass the school gates. 13. The table below shows the weight 40 children under the age of one visiting the health centre. Weight w kg Frequency 1.5 < < < < < < Calculate an estimate for the mean weight of this group of children. S J Cooper 9

11 14. The table below shows the height of 30 boys in year10. Height h cm Frequency 100 h < h < h < h < h < h < h < Calculate an estimate for the mean height of the 30 boys in year Harriet records the time it takes her to travel home from school. The table below shows her records for the last 20 days. Time t minutes Frequency 10 < < < < < 20 1 Calculate an estimate for the mean length of time taken for Harriet to travel home from school. S J Cooper 10

12 Handling data (7) Histograms 1. The table below shows the marks obtained by 120 students sitting an English exam. Marks Frequency a) Represent this data on a histogram. b) Obtain an estimate for the mean score for these 120 students. 2. The histogram drawn opposite represents the Test results of boys in a particular year group. a) How many boys have a test result between 5 and 20? b) What is the greatest possible test mark for a boy in this year group? c) How many boys are there in this year group? frequency The table below shows the distribution of the lengths of time a group Test of Score students spent on their mobile phones one Monday evening. Time (t minutes) 0 t t t t t 75 frequency a) How many students are there in this distribution? b) Represent these findings in a histogram. 4. A local school conducted a survey of the speeds of cars that passed the school gates and obtained the following results. Speed (mph) Number of cars Represent this information in a histogram. S J Cooper 11

13 Frequency Handling Data (8) Frequency Polygons 1. The table below shows the times taken, in minutes, to complete a crossword puzzle for a group of year 9 students. Boys times a) How many boys completed the crossword in less than 15 minutes? b) How many girls completed the puzzle? c) Which gender were the fastest at completing the puzzle? Comment. 2. The table below shows the wages distributed within a factory each week. Represent this data on a frequency polygon. Girls times Time taken Wage Number of employees A group of students from America and England took part in a Mathematics experiment which was assessed by means of a test. The results are shown below. Number of Students Marks English American Represent the information on one frequency polygon Comment on the findings. S J Cooper 12

14 Handling Data (9) Cumulative Frequency curve 1. The table below shows the wages distributed within a factory each week. Represent this data on a cumulative frequency curve. Wage Number of employees The curve below represents the cumulative frequency for the heights of 50 students at a sixth form. Frequency Height (cm) a) From the curve obtain an estimate for the number of students who i. had a height lower than 164cm ii. had a height greater than 180cm. b) Estimate the height of the 25 th student. 3. A group of students from America and England took part in a Mathematics experiment which was assessed by means of a test. The results are shown below. Number of Students Marks English American Draw two separate cumulative frequency curves for this set of data. S J Cooper 13

15 Handling Data (10) Stem & Leaf diagrams 1. The number of cars passing a bridge over a 5 minute interval was recorded over twenty days. The results are shown below. Represent the information in a Stem and leaf diagram. 10, 5, 15, 9, 31, 24, 21, 8, 37, 51 33, 27, 11, 7, 40, 36, 54, 17, 36, The students in a year 8 class sat a mock SATS paper marked out of 60 and obtained the results below. 16, 54, 44, 23, 37, 10, 41, 58, 31, 24, 22 34, 11, 8, 25, 26, 40, 52, 16, 34, 39, 32 21, 9, 10 a. Represent the data in a stem and leaf diagram. b. How many students are their in this class? 3. The stem and leaf diagram below represents the heights of workers in a small supermarket a) How many people are employed at this supermarket? b) What is the height of the 12 th worker, when taken in order of height, starting with the smallest? c) What is the height of the middle person? 4. In a recent cricket match the number of runs scored by each player was recorded as below. Team A Team B a) Represent this information on a double sided stem and leaf diagram b) Which team is more consistent? S J Cooper 14

16 Handling Data (11) Median & Quartiles 1. For each of the following find the median, the Lower Quartile and the Upper Quartile. a) b) c) d) e) With the next five be careful to start with f) 21, 95, 37, 86, 93, 34, 90, 87 g) 22, 48, 77, 94, 56, 77, 29, 98, 92, 59, 93, 88 h) 3, 2, 10, 9, 5, 7, 8 i) 25, 86, 48, 79, 61 j) 589, 204, 796, 508, 730, 902, 81, The table below shows the results of the marks obtained by 8AF in a recent times tables test Mark Frequency Work out the median and quartiles for this set of results. 3. a) Write down five numbers which have a median of 7 b) Write down eight numbers which have a median of The stem and Leaf diagram below represents the marks obtained in a test marked out of a) Obtain the median for this data. b) Obtain the Inter-quartile range for these results. S J Cooper 15

17 Handling Data (12) Box plots 1. For each of the following i) Work out the median, the lower quartile and the upper quartile. ii) Draw a box plot to represent the distribution a) 6, 8, 1, 2, 5, 6, 3, 3, 6, 7, 0 b) 8, 2, 8, 0, 8, 1, 6, 9, 1 c) 87, 23, 77, 25, 16, 52, 8, For the stem and leaf drawn below a. Work out the median, the lower quartile and the upper quartile. b. Draw a box plot to represent the distribution A die is rolled 50 times and the results below show the score obtained per throw. Draw a box plot for this data. Score Frequency The stem and leaf drawn below represents the scores obtained in a test marked out of 50. a. Work out the median, the lower quartile and the upper quartile. b. Draw a box plot to represent the distribution S J Cooper 16

18 Handling Data (13) Cumulative Frequency curve 4. The curve below represents the cumulative frequency for the marks in a test of 40 students at a sixth form. Frequency Mark a) From the curve obtain an estimate for the i. median ii. lower and upper quartiles. b) Work out the interquartile range. 5. The table below shows the masses of 90 girls in at a school. Mass (kg) Frequency Cumulative Frequency a. Copy and complete the table of cumulative frequencies b. Draw the cumulative frequency curve for this set of data c. From the curve find an estimate for i. The median ii. The interquartile range S J Cooper 17

19 6. The table below shows the wages of staff in a local supermarket Wage ( ) Frequency Cumulative Frequency a. Copy and complete the table of cumulative frequencies b. Draw the cumulative frequency curve for this set of data c. From the curve find an estimate for i. The median ii. The interquartile range 7. The table below shows the length of telephone calls taken at the NHS hotline during one day in March. Length of call (t mins) 0 t <15 15 t <30 30 t <45 45 t <60 60 t <75 number of callers a. Draw the cumulative frequency curve for this set of data b. From the curve find an estimate for i. The median ii. The interquartile range 8. Class 8HF sat a Science test marked out of 60 and the marks were placed in the table below. Score x marks 0 x <10 10 x <20 20 x <30 30 x <40 40 x <50 50 x <60 number of Students a. How many pupils are there in 8HF? b. Draw the cumulative frequency curve for this set of data c. From the curve find an estimate for i. The median ii. The interquartile range S J Cooper 18

20 Handling Data (14) Scatter diagrams 1. The table below shows the marks obtained by eight students who sat a test in History and Mathematics. History Mathematics a) Plot the points on a scatter diagram b) Draw onto the diagram a line of best fit c) What type of correlation best describes the set of data. d) Ali was absent for his History test but scored 31 marks in his mathematics test. What score would you predict for him in History? 2. The table below shows the midday temperature and the number of ice creams sold on the front of Briton. Midday Temperature Number of ice creams sold a) Draw a scatter diagram to represent this data. b) What type of correlation is there between the midday temperature and the number of ice creams sold? c) Draw a line of best fit for this scatter diagram 3. The table below shows the time spent watching TV and the time spent on Homework for 12 students. All times are given in minutes. Time spent watching TV Time spent on Homework a) Draw a scatter diagram to represent this data. b) What type of correlation is there between the time spent watching TV and the time spent on Homework? c) Draw a line of best fit for this scatter diagram d) Barbara spent 100 minutes watching TV last night. Estimate the length of time spent on her homework. e) Calvin Spent 50 minutes on his French homework the other night. How long was he likely to have spent watching TV? S J Cooper 19

21 4. The table below shows the house prices of several houses around Britain in the year 2000 and the same houses in All prices given to the nearest thousand pounds. House price in 2000 House price in a) Draw a scatter diagram to represent this data. b) Is there any correlation between the house prices in the year 2000 and in 2007? c) Draw a line of best fit for this scatter diagram d) Darren has a house valued at in the year What would have been its estimated value in the year 2000? e) Explain why the graph can not be used to estimate the price of a house in 2007 which was worth 1.5 million in the year 2000? 5. The table below shows the marks obtained by 10 year 11 students who sat two papers for their recent mock exam. Paper one is a non-calculator paper and paper two is a calculator paper. Paper Paper a) Draw a scatter diagram to represent this data. b) Draw a line of best fit for this scatter diagram c) What type of correlation is there here? d) Dillon was absent for paper 1 but scored 27% on paper 2. Estimate her score on paper 1. e) Yasmin was unable to complete paper 2 but managed 64% on the first paper. What score would you have predicted for Yasmin on paper 2. S J Cooper 20

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