PLC Papers Created For:
Probability of dependent events 2 Grade 5 Objective: To calculate the probability of dependent events using tree diagrams. Question 1. A box of chocolates contains 3 plain, 5 milk and 4 white chocolates. Jenni takes two chocolates from the box at random, what is the probability that she chooses a plain chocolate and a white chocolate?... Question 2 (3) (Total 3 marks) There are 10 cakes on a plate. 3 doughnuts 5 lemon slices 2 cupcakes Husnain takes a cake at random and eats it. He then takes at random a second cake. Work out the probability that Husnain takes two different types of cake.... (3) (Total 3 marks)
Question 3 Martin is often late for school. The probability that he oversleeps is 2. 5 If he oversleeps he is late to school 1 of the time. 2 If he gets up on time he is late 1 of the time. 4 (a) Complete the tree diagram. (2) (b) What is the probability that Martin oversleeps and is late for school?... (3) (Total 5 marks) Total /10
Probability of independent events 2 Grade 5 Objective: To calculate the probability of independent events using tree diagrams. Question 1. In a newsagent's shop, the probability that any customer buys a newspaper is 0.6. In the same shop, the probability that any customer buys a magazine is 0.3. (a) Complete the probability tree diagram. (2) (b) Work out the probability that a customer will buy either a newspaper or a magazine but not both.... (3) (Total 5 marks)
Question 2 Yvonne has 10 tulip bulbs in a bag. 7 of the tulip bulbs will grow into red tulips. 3 of the tulip bulbs will grow into yellow tulips. Yvonne takes at random two tulip bulbs from the bag. She plants the bulbs. (a) Complete the probability tree diagram. (2) (b) Work out the probability that at least one of the bulbs will grow into a yellow tulip. Total /10... (3) (Total 5 marks)
Boxplots 2 Grade 6 Objective: Interpret and construct box plots Question 1 Dave recorded the length of time 24 pupils took to solve a puzzle. The table shows information about these times in minutes. (a) Work out the number of teachers with a travel time of 3.4 minutes or more. (2) (b) On the grid, draw a box plot to show the information in the table. (2) (Total 4 marks)
Question 2 Jodie measures the lengths of 120 snakes. The lengths were recorded and displayed in the boxplot below. (a) Use the boxplot to fill in the values in this summary table: (3) (b) Calculate the Inter Quartile Range (IQR). (2) (c) One of the snakes measured at 42cm was actually 44cm, what affect would this change have on the median? Decrease Stay the same Increase (1) (Total 6 marks) Total marks /10
Cumulative Frequency 2 Grade 6 Objective: Construct and interpret cumulative frequency diagrams (for grouped discrete as well as continuous data) Question 1 The table below shows information about the heights of 60 students. (a) On the grid, draw a cumulative frequency graph for the information in the table. (3) 60 50 CUMULATIVE FREQUENCY 40 30 20 10 0 140 150 160 170 180 190 200 TIME, T SECS
(b) Find an estimate (i) for the median,... (ii) for the interquartile range.... (Total 6 marks) (3)
Question 2 The table shows information about the lengths, in seconds, of 40 TV adverts. (a) Complete the cumulative frequency table for this information. (1)
(b) On the grid, draw a cumulative frequency graph for your table. (2) 45 40 35 CUMULATIVE FREQUENCY 30 25 20 15 10 5 0 0 10 20 30 40 50 60 70 TIME, T SECS (c) Use your graph to find an estimate for the median length of these TV adverts. seconds (1) (Total 4 marks) Total marks /10
Histograms with equal class intervals 2 Grade 5 Objective: Construct and Interpret histograms with equal class widths for discrete as well as continuous data. Question 1 The table below shows the length of 50 earthworms found in a patch of soil. Length, cms Frequency 0 < x 2 5 2 < x 4 10 4 < x 6 13 6 < x 8 16 8 < x 10 6 a) Draw a histogram to represent this data below Total 50 9 8 7 6 FREQUENCY DENSITY 5 4 3 2 1 0 0 2 4 6 8 10 Length, cms
(3) b) Estimate the median length of earthworm... (4) Total for question 7 marks Question 2 The Histogram below show the weights of 30 babies when they are born. 12 10 8 FREQUENCY DENSITY 6 4 2 0 0 1 2 3 4 5 6 Weight, kgs
Using the histogram, complete the table below. Weight, kgs Frequency 0 < x 1 4 1 < x 2 6 2 < x 3 11 3 < x 4 5 4 < x 5 4 Total 30 (3) Total for question 3 marks Total /10
Histograms with unequal class widths 2 Grade 6 Objective: data) Construct and interpret a histogram with unequal class widths (for grouped discrete as well as continuous Question 1 The table shows the length of 678 phone calls made at a call centre Time, secs Frequency 0 < x 20 20 20 < x 60 148 60 < x 120 240 120 < x 300 270 Total 678 a) Draw a fully labelled histogram to show the length of the phone calls. (4)
b) Estimate the number of phone calls that lasted more than 2 minutes. (2) Total for question 6 marks
Question 2 The histogram and the frequency table show some information about the height of some students Height, cms Frequency 130 < x 150 100 150 < x 160 120 160 < x 165 75 165 < x 175 60 175 < x 190 45 Total 400
14 12 10 Frequency Density 8 6 4 2 0 100 120 140 160 180 200 Height, cms a) Use the information to complete the histogram b) Use the histogram to complete the table (2) (2) Total for question 4 marks Total /10
Quartiles and Interquartile Range 2 Grade 5 Objective: Interpolate and calculate quartiles and interquartile range. Question 1 The scores of a maths test are shown below: 37 54 43 57 50 68 69 79 80 77 91 53 96 a) Find the Upper Quartile b) Find the Lower Quartile (2) (1) c) Calculate the Inter-Quartile Range (IQR) (1) d) (i) What is a better measure of spread, Range or Interquartile range (ii) Explain your answer (1).... (1) Total for question 6 marks
Question 2 The table below lists the summary data for the weights of 60 snakes in grams. Weights, kg Minimum 454 L. Quartile 622 Median 660 U. Quartile 812 Maximum 987 a) What is the interquartile range b) Another snake is found to weigh 1340 grams, what effect does this have on (2) (i) The range DECREASE / NO CHANGE / INCREASE (delete as appropriate) (ii) The interquartile range DECREASE / NO CHANGE / INCREASE (delete as appropriate) (1) (1) Total for question 4 marks Total / 10
Scatter Diagrams 2 Grade 5 Objective: Interpret scatter graphs by discussing correlation and causation, draw lines of best fit, make predictions, and interpolate and extrapolate apparent trends whilst knowing the dangers of doing so. Question 1 The table below shows the test results of 10 students in Maths and Science. Maths test score 64 48 79 95 72 32 65 84 56 91 Science test score 72 44 82 98 64 31 61 79 52 97 a) Present this data in a scatter diagram. 120 100 80 SCIENCE SCORE 60 40 20 0 0 10 20 30 40 50 60 70 80 90 100 MATHS SCORE (4) b) Another student scores 80 on the maths test. Estimate their result in the science test (2)
c) Describe the relationship between maths scores and the science scores.... (1) Total for question 7 marks
Question 2 The scatter diagrams below show various types of correlation. Join the scatter diagrams to the labels in the boxes. Weaker positive correlation Weaker negative correlation Stronger positive correlation Stronger negative correlation No correlation Total for question 3 marks Total / 10
PLC Papers Created For:
Probability of dependent events 2 Grade 5 SOLUTIONS Objective: To calculate the probability of dependent events using tree diagrams. Question 1. A box of chocolates contains 3 plain, 5 milk and 4 white chocolates. Jenni takes two chocolates from the box at random, what is the probability that she chooses a plain chocolate and a white chocolate? 3 12 x 4 11 or 4 12 x 3 11 3 x 4 + 4 x 3 12 11 12 11 24 132 = 2 11 M1 M1 A1 oe... (3) (Total 3 marks) Question 2 There are 10 cakes on a plate. 3 doughnuts 5 lemon slices 2 cupcakes Husnain takes a cake at random and eats it. He then takes at random a second cake. Work out the probability that Husnain takes two different types of cake. 3 10 x 5 9 or 3 10 x 2 9 or 5 10 x 3 9 or 5 10 x 2 9 or 2 10 x 3 9 or 2 3 x 5 + 3 x 2 + 5 x 3 + 5 x 2 + 2 x 3 + 2 10 9 10 9 10 9 10 9 10 9 62 = 31 90 45 OR 3 10 x 2 9 or 5 10 x 4 9 or 2 10 x 1 9 10 x 5 9 10 x 5 9 1 ( 3 10 x 2 9 + 5 10 x 4 9 + 2 10 x 1 9 ) M1 62 90 = 31 45 M1 M1 A1 oe M1 A1 oe... (3) (Total 3 marks)
Question 3 Martin is often late for school. The probability that he oversleeps is 2. 5 If he oversleeps he is late to school 1 of the time. 2 If he gets up on time he is late 1 of the time. 4 (a) Complete the tree diagram. Probabilities correctly place on diagram (Allow M1 if at least 3 are correct) M2 (2) (b) What is the probability that Martin oversleeps and is late for school? 2 5 x 1 2 2 = 1 10 5 M1 A1 oe... (2) (Total 4 marks) Total /10
Probability of independent events 2 Grade 5 SOLUTIONS Objective: To calculate the probability of independent events using tree diagrams. Question 1. In a newsagent's shop, the probability that any customer buys a newspaper is 0.6. In the same shop, the probability that any customer buys a magazine is 0.3. (a) Complete the probability tree diagram. B2 for 6 correct probabilities in the correct positions, B1 for 2, 3, 4 or 5 correct (2) (b) Work out the probability that a customer will buy either a newspaper or a magazine but not both. 0.6 x 0.7 + 0.4 x 0.3 M1 for 0.6 x 0.7 or 0.4 x 0.3 M1 for 0.6 x 0.7 + 0.4 x 0.3 OR M1 for 0.6 x 0.3 + 0.4 x 0.7 M1 for 1- ( 0.6 x 0.3 + 0.4 x 0.7 ) A1 0.54... (3) (Total 5 marks)
Question 2 Yvonne has 10 tulip bulbs in a bag. 7 of the tulip bulbs will grow into red tulips. 3 of the tulip bulbs will grow into yellow tulips. Yvonne takes at random two tulip bulbs from the bag. She plants the bulbs. (a) Complete the probability tree diagram. B1 for 3/10 on left hand of yellow branch B1 for rest of fractions correct on tree diagram (2) (b) Work out the probability that at least one of the bulbs will grow into a yellow tulip. M1 for 7 10 x 3 9 or 3 10 x 7 9 or 3 10 x 2 9 M1 for 7 10 x 3 9 + 3 10 x 7 9 + 3 10 x 2 9 OR M1 for 7 10 x 6 9 M1 for 1-7 10 x 6 9 A1 48 90 oe... (3) (Total 5 marks) Total /10
Boxplots 2 Grade 6 SOLUTIONS Objective: Interpret and construct box plots Question 1 Dave recorded the length of time 24 pupils took to solve a puzzle. The table shows information about these times in minutes. (a) Work out the number of teachers with a travel time of 3.4 minutes or more. 75% of 24 = (18) M1 (b) On the grid, draw a box plot to show the information in the table. 18 A1 (2) (2) (Total 4 marks)
Question 2 Jodie measures the lengths of 120 snakes. The lengths were recorded and displayed in the boxplot below. (a) Use the boxplot to fill in the values in this summary table: Length, cms Minimum 24 Lower Quartile 38 Median 40 Upper quartile 46 Maximum 54 (3) (b) Calculate the Inter Quartile Range (IQR). 46 38 = (8) M1 8 A1 (2) (c) One of the snakes measured at 42cm was actually 44cm, what affect would this change have on the median? Decrease Stay the same Increase (1) (Total 6 marks) Total marks /10
Cumulative Frequency 2 Grade 6 SOLUTIONS Objective: Construct and interpret cumulative frequency diagrams (for grouped discrete as well as continuous data) Question 1 The table below shows information about the heights of 60 students. (a) On the grid, draw a cumulative frequency graph for the information in the table. (3) 70 60 CUMULATIVE FREQUENCY 50 40 30 20 10 0 140 150 160 170 180 190 200 HEIGHT, CMS
(b) Find an estimate (i) for the median,...172...a1... (ii) for the interquartile range....177 165 = (12)...M1 A1 (Total 6 marks) (3)
Question 2 The table shows information about the lengths, in seconds, of 40 TV adverts. (a) Complete the cumulative frequency table for this information. (1) (b) On the grid, draw a cumulative frequency graph for your table. (2)
CUMULATIVE FREQUENCY 45 40 35 30 25 20 15 10 5 0 0 10 20 30 40 50 60 70 TIME, T SECS (c) Use your graph to find an estimate for the median length of these TV adverts. 37 B1 seconds (1) (Total 4 marks) Total marks /10
Histograms with equal class intervals 2 Grade 5 SOLUTIONS Objective: Construct and Interpret histograms with equal class widths for discrete as well as continuous data. Question 1 The table below shows the length of 50 earthworms found in a patch of soil. Length, cms Frequency Class Width Freq. Density 0 < x 2 5 2 2.5 2 < x 4 10 2 5.0 4 < x 6 13 2 6.5 6 < x 8 16 2 8.0 8 < x 10 6 2 3.0 Total 50 a) Draw a histogram to represent this data below 9 8 7 FREQUENCY DENSITY 6 5 4 3 2 1 0 0 2 4 6 8 10 Length, cms
(3) b) Estimate the median length of earthworm. Location of median = (50+1)/2 = 25.5 th worm Median is in the 4-6 group. 5 + 10 + X = 25.5 so x is 10.5 (into the group) M1 10.5 / 6.5 (freq density and height of class) = 1.615 M1 4 + 1.615 = (5.615 cms) M1 5.615 A1.. (4) Total for question 7 marks Question 2 The Histogram below show the weights of 30 babies when they are born.
12 10 8 FREQUENCY DENSITY 6 4 2 0 0 1 2 3 4 5 6 Weight, kgs Use the histogram to complete the table below. The vertical scale can be any set of numbers but this will affect the f.d. calcs. Weight, kgs Frequency Class Width Freq. Density 0 < x 1 4 1 4.0 1 < x 2 6 1 6.0 2 < x 3 11 1 11.0 3 < x 4 5 1 5.0 4 < x 5 4 1 4.0 Total 30
(3) Total for question 3 marks Total /10
Histograms with unequal class widths 2 Grade 6 SOLUTIONS Objective: data) Construct and interpret a histogram with unequal class widths (for grouped discrete as well as continuous Question 1 The table shows the length of 678 phone calls made at a call centre Time, secs Frequency Class Width Freq. Density 0 < x 20 20 20 1.0 20 < x 60 148 40 3.7 60 < x 120 240 60 4.0 120 < x 300 270 180 1.5 Total 678 a) Draw a fully labelled histogram to show the length of the phone calls. 4.5 4 3.5 3 Frequency Density 2.5 2 1.5 1 0.5 0 0 50 100 150 200 250 300 350 Time, secs (4)
b) Estimate the number of phone calls that lasted more than 2 minutes. 4 minutes = 4 x 60 secs = 240 secs 300 240 = 60 mins 60 x 1.5 = (90 calls) M1 90 calls A1 (2) Total for question 6 marks
Question 2 The histogram and the frequency table show some information about the height of some students Height, cms Frequency Class Width Freq. Density 130 < x 150 100 20 5.0 150 < x 160 120 10 12.0 160 < x 165 75 5 15.0 165 < x 175 60 10 6.0 175 < x 190 45 15 3.0 Total 400 16 14 12 Frequency Density 10 8 6 4 2 0 100 120 140 160 180 200 Height, cms a) Use the information to complete the histogram b) Use the histogram to complete the table (2) (2) Total for question 4 marks Total /10
Quartiles and Interquartile Range 2 Grade 5 SOLUTIONS Objective: Interpolate and calculate quartiles and interquartile range. Question 1 The scores of a maths test are shown below: 37 54 43 57 50 68 69 79 80 77 91 53 96 Minimum 37 L. Quartile 53 Median 68 U. Quartile 79 Maximum 96 a) Find the Upper Quartile Write numbers in order M1 b) Find the Lower Quartile 79.5 A1 (2) 51.5 A1 (1) c) Calculate the Inter-Quartile Range (IQR) 79 53 = (26) M1 28 A1 (1) d) (i) What is a better measure of spread, Range or Interquartile range (ii) Explain your answer Interquartile range A1 (1) Range does not allow for outliers whereas Interquartile only look sat middle chunk of data. C1.. (1) Total for question 6 marks
Question 2 The table below lists the summary data for the weights of 60 snakes in grams. Weights, kg Minimum 454 L. Quartile 622 Median 660 U. Quartile 812 Maximum 987 a) What is the interquartile range 812 622 = (190) M1 b) Another snake is found to weigh 1340 grams, what effect does this have on 190 A1 (2) (i) The range DECREASE / NO CHANGE / INCREASE (delete as appropriate) (ii) The interquartile range DECREASE / NO CHANGE / INCREASE (delete as appropriate) (1) (1) Total for question 4 marks Total / 10
Scatter Diagrams 2 Grade 5 SOLUTIONS Objective: Interpret scatter graphs by discussing correlation and causation, draw lines of best fit, make predictions, and interpolate and extrapolate apparent trends whilst knowing the dangers of doing so. Question 1 The table below shows the test results of 10 students in Maths and Science. Maths test score 64 48 79 95 72 32 65 84 56 91 Science test score 72 44 82 98 64 31 61 79 52 97 a) Present this data in a scatter diagram. 120 100 80 SCIENCE SCORE 60 40 20 0 0 10 20 30 40 50 60 70 80 90 100 MATHS SCORE (4) b) Another student scores 80 on the maths test. Estimate their result in the science test Line of best fit M1 80 A1 (2)
c) Describe the relationship between maths scores and the science scores Positive correlation or The better a student does in science, the better they will do in science. A1.. (1) Total for question 7 marks
Question 2 The scatter diagrams below show various types of correlation. Join the scatter diagrams to the labels in the boxes. Weaker positive correlation Weaker negative correlation Stronger positive correlation Stronger negative correlation No correlation Total for question 3 marks Total / 10