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1 Mathematicsisliketravellingona rollercoaster.sometimesyouron Mathematics ahighothertimesyouronalow.ma keuseofmathsroomswhenyouro Stage 6 nalowandshareyourpracticewit Handling Data hotherswhenonahigh.successwi S J Cooper llfollowifyoufollowthisadvice.th isbooklooksatalgebraatahighlev el.yournotexpectedtoknowitallb utyouareexpectedtoalwaysdoyo urbest.thebookconsistsofappro ximatelythirtyworksheets.henc eoneaweekwilltakeatleastyeart entocomplete.oncecompletedwe canlookatpapersandironoutany creases.askandhelpwillalwaysb eprovidguidanceisablessingtoh aveasisanendgoaltoaspiretowar ds.enjoyhavefunandworkwell.

2 Handling data(1) 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.

3 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?

4 Handling Data (2) Cumulative Frequency 1. The table shows the cricket scores during one season. Score Frequency a) Copy and complete the table to show the cumulative frequencies. b) Draw the cumulative frequency curve. 2. The table below shows the results in an English test. Mark Frequency a) Draw the table of cumulative frequencies. b) Draw the cumulative frequency curve. c) Give that 54% was the lowest mark for a pass, estimate how many pupils scored less than 54% 3. 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 0 s 5 5 s s s s s 30 (s) Frequency Draw the cumulative frequency curve which represents these results.

5 4. The table below is the cumulative frequency table for the results in a Mathematics test. Percentage mark Cumulative frequency a) Represent this information on a cumulative frequency curve. b) Estimate how many students scored over 52%? 5. John is fed up of always being the one who answers the telephone in his household and so decides to leave it to ring until someone else answered it. In doing this he noted down the number of rings before the telephone was answered. The cumulative frequency table gives the number of rings before it was answered. Number of rings Cumulative frequency a) Represent this information on a cumulative frequency curve. b) On how many occasions did the telephone ring for more that 12 times before being answered? 6. The marks out of fifty obtained by a year 11 set on its end of term test were as follows: Number of rings Cumulative frequency a) Represent this information on a cumulative frequency curve. b) Estimate the number of students who scored over 35 marks in this test.

6 Handling Data (3) Box plots 1. For each of the following a. work out the median, the lower quartile and the upper quartile. b. 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.

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. 1. 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

8 6. 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 a. Draw up a cumulative frequency table and draw the cumulative frequency curve. b. Use the graph to estimate the median and find the inter-quartile range. c. What is the probability that an applicant drawn at random i. belongs to the group ? ii. is 33 years or younger? 7. The time t (in seconds) of 180 local telephone calls is recorded below. Time (t seconds) 0 t t t t t 120 Frequency a. Draw a cumulative frequency diagram to illustrate this information. b. Use the diagram to obtain the median and inter-quartile range for the length of telephone calls. c. 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 375.

9 Handling Data(5) 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.

10 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 ( x thousand) Number of sales 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.

11 Handling Data (6) 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 Number of students 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 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 0 t 5 5 t t t t 25 mins) 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 Frequency 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.

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

14 Handling Data (7) Mean for grouped data 1. 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. 2. The scores of a cricketer in one season are given in the table below: Score Frequency Calculate an estimate for the mean score. 3. 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 daytrippers visiting Blackpool by coach. She obtains the following information from 130 coach drivers. Distance travelled (d 0 d d d d d 160 miles) Number of coaches Estimate the mean distance travelled by the 130 coach drivers.

15 5. 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. 6. The table below gives the distribution about the population of Males in England and Wales in Population (millions) Age (years) Males Calculate an estimate of the mean age for males in The histogram drawn 40 opposite represents the heights of girls in a particular year group a) How many girls have a height between 160 and 170 cm? b) Estimate the mean height of these girls Heights (cm)

16 Handling data(8) Probability: 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.

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