Thomas Whitham Sixth Form. Higher Tier

Transcription

1 Thomas Whitham Sixth Form YEAR 10 Higher Tier HANDLING DATA Probability Tree diagrams & Sample spaces Statistical Graphs inc. Histograms Scatter diagrams Mean, Mode & Median Sampling

2 Handling Data (1) Mean Mode & Median 1. Find the mean and range for each of the following sets of numbers: a) 51, 88, 95, 92, 14, 42, 11 b) 75, 23, 27, 79, 10, 89, 70, 64, 1, 25 c) 37, 47, 7, 93, 10, 72, 78, 0 d) 0.42, 13.91, 26.26, 99.45, e) 77.2, 20.3, 58.2, 2.5, 39.3, 12.7, 66.8, Over the past eight weeks Najid and David have compared their weekly maths tests. Najid s last eight test marks were 54, 43, 57, 19, 79, 45, 49 and 66. While David s last eight marks were 51, 49, 56, 38, 69, 47, 52 and 63. a) Which student is the better? Give a reason for your answer. b) Which student is the more consistent? Give a reason for your answer. 3. Over the past two weeks James makes a note of how long it takes him to travel to school in a morning. His times, to the nearest minute, are shown below. 43, 37, 26, 49, 32, 51, 19, 35, 24, 22 What is his average time taken in travelling to school? 4. Whilst playing a game of snooker Alex recorded the highest breaks he made in each game. His results are shown below. 47, 58, 10, 87, 59, 104, 69 What is his average highest break from these results? 5. Over the past term Jenny has been given tests in both Science and Maths. Her average mark for six Science tests was While in Maths her average mark for four tests was Work out her mean mark for all ten tests. 6. Work out the mode for each of the following set of numbers: a) 1, 5, 7, 3, 1, 8 b) 5, 6, 1, 8, 2, 6, 2, 4, 6 c) 18, 13, 12, 9, 16, 12, 14, 9, 10, 12, 11 1

3 7. The table below shows the results when a die is through fifty times. Work out the modal score. Score Frequency Total In a fund raising competition at school, pupils were sponsored for the number of spellings they had correct. The results are shown in the table below. No. of spellings (s) 0 s 5 5 s s s s s 30 Frequency What is the modal class for this data? 9. For each of the following work out (i) the median (ii) the lower quartile (iii) the upper quartile. a) 1, 9, 9, 1, 6, 8, 6, 7, 4, 5, 5 b) 63, 39, 37, 72, 63, 77, 66, 84, 2, 16 c) 1.1, 5.2, 6.0, 0.5, 3.6, 6.3, 5.6, 0.4, 2.1, 0.5, 7.7 d) 22, 48, 77, 94, 56, 77, 29, 98, 92, 59, 93, 88 e) 7.4, 4.4, 9.9, 9.6, 5.3, 4.9, 4.2, 9.2, 2.3, 0.7, 1.9, 3.7 f) 589, 204, 796, 508, 730, 902, 81, The table below shows the results of the marks obtained by 7BF in a recent times tables test Mark Frequency Work out the median and quartiles for this set of results. 11. Six coins are tossed together 50 times and the results below show the number of heads obtained per throw. Work out the median and quartiles for this data. Number of heads Frequency

4 Handling Data (2) Box plots 1. For each of the following (i) (ii) work out the median, the lower quartile and the upper quartile. 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, 80 d) 48, 45, 28, 4, 24, 75, 16, 13, 3, 58, 57 e) 0.4, 0.4, 0.1, 0.2, 0.7, 1.0 f) 0.66, 0.17, 0.49, 0.72, 0.23, 0.75, 0.92, 0.34 g) 0.4, 3.4, 2.4, 6.7, 8.3, 8.4, 6.5, 7.6, 9.0, 7.0 h) 73, 79, 36, 69, 9, 92, 8, 82, 51, 57, 42, The table below shows the results of the marks obtained by 8BF in a recent times tables test Mark Frequency a) Work out the median and quartiles for this set of results. b) Draw a box and whisker diagram to illustrate this data. 3. Six coins are tossed together 50 times and the results below show the number of heads obtained per throw. Draw a box plot for this data. Number of heads Frequency A fitness club noted the number of press ups its members could achieve in a 30 second interval and produced the following results. 10, 17, 32, 18, 6, 21, 11, 6, 26, 13, 14, 9, 16, 15, 12, 20, 19, 7, 8, 7, 14, 11, 22, 24, 8, 16, 18, 12, 22, 24, 10 Represent these findings in a box plot. 3

5 Handling Data (3) Mean 5. A school conducted a survey on the number of sandwiches parents made for their children each day. The results are shown in the table below. Number of sandwiches Frequency Calculate the mean number of sandwiches per pupil. 6. Henrietta is Farmer Giles favourite hen because of the number of eggs she lays. The table below shows the distribution of the number of eggs laid by Henrietta over a period of time. Work out the average number of eggs Henrietta lays. Number of eggs frequency The vertical line graph shows the results to a survey on the shoe size of a group of pupils. a) Which shoe size is most common? b) Work out the mean shoe size for this group of students Frequency Shoe size 4. The table below shows the number of peas per pod some 55 selected pods. Number of peas Frequency Calculate the mean number of peas per pod. 4

6 5. The frequency table below represents the number of minutes female members of staff spend on the telephone during breaktime. (Time given to the nearest minute). Duration (mins) Frequency Estimate the mean amount of time spent on the phone. 6. The scores of a cricketer in one season are given in the table below: Score Frequency Calculate an estimate for the mean score. 7. The frequency table below represents the age distribution of members of staff at Castlerock High school. Calculate an estimate for their mean age. Age of staff (n years) 21 n n n n n 71 Number of staff A pupil working on a GCSE project investigates the distances travelled by day-trippers visiting Blackpool by coach. She obtains the following information from 130 coach drivers. Distance travelled (d miles) 0 d d d d d 160 Number of coaches Estimate the mean distance travelled by the 130 coach drivers. 9. The times taken by 150 applicants to complete a test are shown in the table below. Time (t minutes) 0 t t t d 50 Number of applicants Calculate an estimate of the mean time taken for the test. 5

7 Handling Data (4) Cumulative Frequency For each of the following questions (i) Draw the cumulative frequency curve (ii) find the median value, (iii) find the inter-quartile range. 8. The table below shows the frequency table for the marks obtained in a science examination by 120 students. Mark Frequency The table shows the marks obtained in a mathematics examination by the same 120 students. Mark Frequency Darts are thrown randomly at a dart board and the scores are recorded in the table below. Score off 1 dart Frequency In a year group containing 90 boys their masses were obtained and placed in the table below. Mass (kg) Frequency The table below gives the distribution of heights, to the nearest centimetre, of 66 fourteen year olds. Height (cm) Frequency The ages, in complete years of forty applicants for a teaching post are given in the table below: Age (n years) 20 n n n n n 45 Frequency (i) Draw up a cumulative frequency table and draw the cumulative frequency curve. (ii) Use the graph to estimate the median and find the inter-quartile range. (iii) What is the probability that an applicant drawn at random i. belongs to the group ? ii. is 33 years or younger? 6

8 14. The time t (in seconds) of 180 local telephone calls is recorded below. Time (t seconds) 0 t t t t t 120 Frequency (i) Draw a cumulative frequency diagram to illustrate this information. (ii) Use the diagram to obtain the median and inter-quartile range for the length of telephone calls. (iii) It is decided to increase telephone charges for calls of over 45 seconds duration. Estimate the number of telephone calls in this sample which will be effected. 8. A travel firm have a brochure advertising holidays; the prices are varied depending on the destination. The table below shows the grouped amounts people paid for their holiday over the month of February. Cost per person ( ) Number of holidays a) Draw a cumulative frequency diagram to illustrate this information. b) Use the diagram to obtain the median and inter-quartile range for the cost per person in this brochure. c) Estimate the number of holidays which cost more than

9 Handling Data (5) Moving Averages 1. Work out the three point moving average for each of the following data a) 12, 15, 9, 21, 21, 18 b) 4, 8, 15, 19, 17, 6 c) 1.3, 1.7, 2.4, 2.8, 2.3, 3.3 d) 87, 75, 66, 72, 81, 84, 78 e) 13.5, 20.6, 14.5, 19.8, 21.8, 24.1, Johns last eight maths test results are listed below 32, 54, 28, 62, 54, 52, 47, 60 a) Find the three point moving average for his results. b) Has his understanding in the subject improved? Give a reason for your answer. 3. At the end of each month Mr Smith receives a building society statement which tells him how much he has in his account. The table below shows the amount in his account at the end of each month for the finance year April to March. Month April May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. March Balance, Copy and complete the table below to find the three point moving average. Month April May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. March Average Balance, During the ten school days, before the summer holidays last year, David recorded the outside temperature at lunchtime and placed the results in the table below. Day Mon. Tues. Weds. Thurs. Fri. Mon. Tues. Weds. Thurs. Fri. Temp C Copy and complete the table below to find the five point moving average. Day Mon. Tues. Weds. Thurs. Fri. Mon. Tues. Weds. Thurs. Fri. Average Temp C 8

10 5. A small business makes both Jeans and Cords. Their manager takes a note of the number of each made over several weeks. Represent this information on a time series graph. Week ending 1 Aug 8 Aug 15 Aug 22 Aug 29 Aug 5 Sept 12 Sept 19 Sept 26 Sept Number of Jeans Number of Cords Copy and complete the table below to find the three point moving average, giving answers where needed correct to one decimal place. Week ending 1 Aug 8 Aug 15 Aug 22 Aug 29 Aug 5 Sept 12 Sept 19 Sept 26 Sept Average Number of Jeans Average Number of Cords 6. The share price of an ordinary Share is noted at the end of each day for the last seven days of trading on the stock market, as shown on the graph drawn opposite. a) Represent this data in a table b) Work out the three point moving average for these share prices. Value, ( ) M T W Th F M T Day of the week 9

11 Handling Data (6) Scatter diagrams 1. Fifteen students were given two tests, one in mathematics the other in English. The percentages they obtained are shown in the table below. Mathematics Mark English Mark a) Draw a scatter diagram to represent this data. b) What type of correlation is there between the marks obtained in mathematics and those obtained in English? c) Draw a line of best fit for this scatter diagram d) Peter obtained 37% in mathematics but was absent for the English test. What mark would you expect him to obtain in English? e) Which test did the students find more difficult? Give a reason for your answer. 2. A group of ten fathers and their fifteen year-old sons were selected at random and their heights measured, correct to the nearest centimetre. The table below shows their findings: Height of father Height of son a) Draw a scatter diagram to represent this data. b) Draw a line of best fit through the points. c) What type of correlation is there between the heights of fathers and the heights of their sons? d) Najid is 2 years old and his father has a height of 1.7 metres. What height would you predict for Najid when he is 15 years old? 3. A class of twelve students were given a test. At the end of the test the students were asked to count the number of items in their pockets. The table below shows both sets of results: Test mark Items in pocket a) Draw a scatter diagram which represents this data. b) What conclusion could you make about the correlation between number of items in their pockets? 10

12 4. In a particular school 15 students sat both a French exam and a History exam. The results are as follows: History Mark French mark a) Draw a scatter diagram to represent this data. b) What type of correlation is there between the marks obtained in French and the marks obtained in History? c) A newcomer to the school sat the History exam and this was used to predict his French mark. If her History mark was 56% what mark would she be given for her French exam? d) Adrian Obtained 31% in his French exam but truanted his history exam. What mark could his class teacher use for his history exam? 5. Sketch a scatter diagram which best describes the relationship between the following variables: a) Number of hour s sunshine in a day at Blackpool and the number of Ice creams sold on the beach. b) Number of hours rain in one day and the number of ice creams sold that day. c) Number of hours sunshine in a given day and the number of hours homework set that day. d) Number of hours sunshine in a day and the number of hours sleep in that day. 11

13 Number of students Handling Data (7) Frequency Polygons 1. The students of a private school were all asked to record the length of time it took them to get to school in a morning. The results were placed in the table below. Length of travel (mins) Number of students Draw the frequency polygon to represent this information. 2. The frequency polygon drawn represents last years SATS results for the students at Habergham in Mathematics and English Mathematics English Level obtained a) Which subject did the students find the easiest? Give a reason for your answer. b) How many students obtained a level 8 in (i) English (ii) Maths? c) How many students obtained a level 5 or more in (i) English (ii) Maths? 12

14 Frequency 3. Following a recent telephone bill Jason decides to note down the approximate length of all telephone calls in his household for the month of May. His results are placed in the table below. Length of call (t mins) 0 t 5 5 t t t t 25 Frequency a) How many telephone calls were made in the month of May? b) Draw a frequency polygon to represent these results. 4. The frequency polygon drawn represents the weekly pocket money given to two different classes in a particular primary school Class A Class B Pocket money a) How many pupils in class A received between 5 and 10 pocket money each week? b) How many pupils are there in class B? c) If one class is older than the other, which class has the older pupils? Give a reason for your answer. 5. The scores of a cricketer in one season are given in the table below: Score Frequency a) How many runs did this cricketer make this season? b) Represent this data on a frequency polygon. 13

15 6. The table below gives the distribution about the population of Females in England and Wales in Age (years) Population (millions) Females Represent this information on a frequency polygon

16 Handling Data (8) Stem & Leaf diagrams 1. A class of 50 primary students sat a aural test in preparation for their SATS and the scores out of 40 are given below Represent this information in a stem and leaf diagram. 2. The heights of a group of 30 boys were recorded by their class teacher. (all measurements in cm) Represent these results in a stem and leaf diagram. 3. In a recent mathematics test a competition between boys and girls gave the following results. Boys Girls Represent this information on a double sided stem and leaf diagram. Who won the competition the boys or the girls. Give a reason for your answer. 15

17 Handling Data (9) other charts 1. The Bar chart below represents the favourite subjects of class 6B History Maths English Science French Art (a) How many children chose Maths as their favourite subject? (b) Which subject was the most popular? (c) Which subject was the least popular? (d) Represent this same information in a Pie chart. 2. A small class of 25 pupils were asked Where they spent their summer holidays. The results are shown in the table below. Hoiday frequency Cornwall 9 Greece 3 Spain 6 USA 7 Illustrate these results in a pie chart. 3. In a recent survey on the types of car on the school car park the pie chart below was constructed. Type of car on the school car park BMW VOLVO TOYOTA FORD NISSAN 16

18 a) What size angle represents Nissan? b) What percentage of the car park was Volvo? c) If there were ninety cars in total on the car park, how many cars were Ford s? 4. The local School did a survey with its year 6 pupils, on the number people living in their house. The results are shown in the table below. Number of People in their house a) Represent the data in a tally chart b) Draw a vertical line to represent these results. c) How many children had three others living in their house? d) How many families had five living in their house? Frequency 5. The vertical line graph shows the results to a Test out of ten given to a group of pupils. c) Which mark is the most common? d) How many pupils sat this test? e) What is the probability of a pupil selected at random scored 8 or more? This frequency table shows the results of a survey on children s opinions about the quality of their maths homework Test result Opinion Very Good Good Satisfactory Poor Very Poor Frequency a) How many people were asked their opinions on their maths homework? b) Draw a bar chart to show this information. 17

19 Handling Data (10) Histograms 1. A travel firm have a brochure advertising holidays; the prices are varied depending on the destination. The table below shows the grouped amounts people paid for their holiday over the month of February. Cost per person ( ) Number of holidays Represent this information in a histogram. 2. The table below shows the distribution of the ages of people visiting a local dentist in the month of March. Illustrate this information In a histogram. Age (n) 0 n 5 5 n n n n 80 frequency The table below represents the distribution of heights of boys in a particular social club. Heights (cm) frequency a) How many boys are there in this distribution? b) Represent these findings in a histogram. 4. The histogram drawn opposite represents the heights of girls in a particular year group Frequency density a) How many girls have a height between 155 and 160cm? b) What is the greatest possible height for a girl in this year group? c) How many girls are there in this year group? Heights (cm) 18

20 Handling Data(11) Histograms 1. The heights of 100 children entering a school were measured and the following table was produced. Height (cm) Frequency 70 < h < h < h < h < h < h a) Calculate the frequency density for each class interval. b) Draw, on graph paper, a histogram to represent this data. c) Estimate the median height 2. The following table summarises the distance, x thousand miles, covered in a particular month by 100 lorry drivers in a large organisation. Distance Frequency 1 < x < x < x < x < x < x 10 6 a) Using graph paper, draw a histogram to illustrate these data. b) Estimate the median of these data c) Estimate the number of lorry drivers who travel over 6800 miles in a particular month. 19

21 3. The following table shows the number of house sales in the various price ranges made by an Estate agent during a particular year. Price range Number of sales ( x thousand) 25 < x < x < x < x < x < x a) Using graph paper, draw a histogram for this data b) For this data, estimate (i) the median (ii) the interquartile range. 4. The 120 students at a sporting academy are required to take a fitness test. Part of the test involves covering an obstacle course. The time, x minutes, taken by each student to cover the course is recorded and the results are summarised below. Time (x minutes) Frequency 20 < x < x < x < x 38 7 a) Draw a histogram to represent this data b) Estimate the median c) Students covering the course in less than 25 minutes pass this part of the test. Estimate the percentage of students passing this part of the test. 20

22 Handling data(12) Sampling 1. Explain briefly the difference between a CENSUS and a SAMPLE. 2. Give four possible reasons why it may be preferable to take a sample rather than take a census. 3. A factory has 500 employees, each one having a works number for the purposes of a survey. A sample of 25 is picked from the workforce. (a) What do you understand by a simple random sample? (b) Describe two methods whereby a simple random sample of 25 can be chosen. (c) Explain why a random sample might not be representative of population. Illustrate if you can in the context of a random sample of 25 from a workforce of 500 (males, females, skilled, unskilled, managers,...hint, hint!) (d) What do you understand by stratified sampling? Illustrate by reference to possible strata in this factory. Comment on aspects of randomness in this process. 4. In a small village, the population is divided by age group as follows: AGE (YEARS) FREQUENCY A sample of 40, stratified according to age is to be taken. How many should be chosen from each age group? For the age group 45-64, explain how you would select people for the sample. 5. The table below shows the number of pupils in each year group of a new school in Burnley Year Group Number of pupils Alan takes a stratified sample of 60 pupils from this school. Work out how many pupils he must select from each year group. 21

23 6. The Local authority wants to conduct a survey about the methods of transport used by students and a local college and found the following results from the college register: Transport Car Bus Walk Taxis Bike Train No. of people The authority wishes to conduct a stratisfied sample of 30 students from the list above. How many students from each category should they select? 7. At a large U.S university, students are classified as follows: HOUSING NUMBER OF STUDENTS CAMPUS DORMITORY 2100 FRATERNITY HOUSE 720 PRIVATE RESIDENCE 3400 In a survey on accommodation, a stratified sample of 200 students is to be taken. What numbers should be taken from each stratum? 22

24 Handling data (13) Probability of a single event 1. A unbiased die is thrown. What is the probability of obtaining (a) a score of 3 (b) a prime number (c) a factor of 10? 2. The numbers on a set of 28 dominoes are as follows: One domino is selected at random from its box. What is the probability of selecting (a) a double 5 (b) a double (c) at least one three (d) one with a total of six dots on it? 3. If a raffle sold 500 tickets of which I bought 40, what is the probability that a) I win first prize b) I win second prize given that someone else won first prize? 4. A computer game for three players is played by Alan, Carol and Emily. The probability that Alan wins the game is 0.3, whereas the 3 probability that Carol will win is. From this information what is the 8 probability that Emily will win the game. Who is most likely to win any given game? 23

25 5. Two ordinary dice are rolled. The score being the total on the two dice. a) Draw a sample space diagram to show all possible outcomes. b) What is the probability of obtaining a score of (i) 5 (ii) at least 8 (iii) Prime? 6. A game consists of two piles of cards each pile numbered from 1 to 5. One card is selected from the top of each pile and 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 Pile Second Pile d) What is the probability that a player will obtain a score of (i) 3 (ii) 4 (iii) 5? 24

26 Handling data(14) Two events 1. Two dice are rolled. Three events are defined as A = the score is a double B = the total score is a four C = the total score is seven Find the probability of the following events a) P(A), P(B), P(C), P(A or B), P(A or C), P(A and B), P(A and C), P(B and C) b) Explain why P(A or C) = P(A) + P(C) But P(A or B) P(A) + P(B) Which of these events are independent? 2. Three coins are tossed what is the probability of obtaining (i) two heads (ii) at least two heads 3. A fair die is thrown together with a biased die. The probabilities of each score on the biased die are given as follows 1 P( 6) 5 P( 1) P(2) P(3) P(4) 1 6 P( 5) Find the probability of obtaining a score (i) equal to (ii) equal to 9 4. What is the probability of throwing a coin five times and obtaining (i) 5 heads (ii) 3 heads and 2 tails 5. The letters of the word MATHEMATICS are written onto identical pieces of card and placed in a bag. A card is selected from the bag replaced and then a second card is selected. Find the probability of obtaining (a) the letter M twice (b) a vowel at least once. 25

27 Handling Data (15) Tree diagrams 1. A Sam goes on holiday every summer and the probability that they go to Margate is st year 2 nd year Margate Margate Elsewhere Margate Elsewhere Copy and complete the tree diagram to find The probability that (a) they go to Margate for the next two years (b) they don t go to Margate in the next two years (c) they go to Margate this summer but not next summer. Elsewhere 2. Four out of every ten students study Physics at A level. (a) What is the probability that a student will not study Physics at A level? (b) Two students are selected at random. Draw a tree diagram to work out the probability that (i) both students study Physics at A level (ii) one student studies Physics at A level. 3. A multiple choice test consists of five possible answers for each question, of which only one answer is correct. Both Helen and Paul could not answer the first question and so decided to choose an answer at random. (a) What is the probability that Helen answers the question correctly? (b) What is the probability that Paul answered the question incorrectly? (c) Copy and complete the tree diagram below: Helen s answer Paul s Answer Correct Correct Incorrect Correct Incorrect Incorrect 26

28 (d) Using the tree diagram what is the probability that (i) Helen only answered the question correctly, (ii) Both answered the question correctly, (iii) Only one of them answered the question correctly? 4. The probability of rain before I leave for school is 5 3. If it is raining the probability that I will have my 2 1 umbrella in the car is. If it is not raining the probability of having my umbrella in the car is only. 3 5 Draw up a tree diagram and use it to find the probability that a) it is raining and the umbrella is in the car, b) its not raining and I don t have my umbrella in the car c) I don t have my umbrella yet its raining. 5. On my way to work in a morning I must pass through two sets of traffic lights. The probability that the 2 1 first set is at green is, whereas the probability that the second set is on green is only. 3 4 (a) What is the probability that the first set of lights will be at red? (b) Copy and complete the tree diagram below (c) What is the probability that both sets of light will be on red? (d) What is the probability that I don t get stopped at any of the lights tomorrow morning? (e) What is the probability that I get held up on at least one set of traffic lights? 1 st set 2 nd set