Amara surveyed 30 pupils in her class to find out how many pets each pupil owns. The frequency table shows her results.
(a)Write down the mode.(1)
(b)Work out the range of the number of pets.(1)
(Total for Question 1 is 2 marks)
2
The frequency table shows the number of pets owned by the same 30 pupils as in Question 1.
(Total for Question 2 is 2 marks)
3
The frequency table shows the number of pets owned by the same 30 pupils as in Questions 1 and 2.
(Total for Question 3 is 3 marks)
4
A hockey team's coach recorded the number of goals the team scored in each of 20 matches last season. The frequency table shows the results.
(a)Write down the mode.(1)
(b)Find the median number of goals scored.(2)
(c)Calculate the mean number of goals scored per match. Give your answer correct to 2 decimal places.(3)
(Total for Question 4 is 6 marks)
5
A researcher asked 25 students how many text messages each of them sent during a one hour period. The frequency table shows the results.
(Total for Question 5 is 3 marks)
6
40 runners took part in a charity fun run. Their finishing times were grouped and recorded in the frequency table below.
(a)Write down the modal class.(1)
(b)Work out how many runners completed the fun run in under 40 seconds.(1)
(c)Find the class interval that contains the median time.(2)
(Total for Question 6 is 4 marks)
7
Using the fun run data from Question 6, calculate an estimate for the mean finishing time.
(Total for Question 7 is 4 marks)
8
A transport survey asked 50 people how far, d km, they travel to work each day. The grouped frequency table shows the results.
(a)Calculate an estimate for the mean distance travelled to work. Give your answer correct to 2 decimal places.(4)
(b)Give a reason why your answer to part (a) is only an estimate of the mean.(1)
(Total for Question 8 is 5 marks)
9
A cafe manager surveyed 50 customers about how many cups of tea each one drinks per day. One frequency value, y, is missing from the table. The total number of customers surveyed was 50.
(Total for Question 9 is 2 marks)
10
A weather station recorded the minimum temperature, in degrees Celsius, on each of 20 days in January in Leeds. The frequency table shows the results.
(Total for Question 10 is 3 marks)
11
Two classes, A and B, sat the same mathematics test, marked out of 10. The frequency table shows the results for Class A.
(Total for Question 11 is 3 marks)
12
The frequency table shows the results for Class B in the same test as Question 11 (also marked out of 10, also 20 pupils).
(a)Calculate the mean score for Class B. Give your answer correct to 2 decimal places.(3)
(b)Find the range of scores for Class A and the range of scores for Class B.(2)
(Total for Question 12 is 5 marks)
13
The mean and range for Class A and Class B are summarised below. Use these to compare the two classes' test results.
(Total for Question 13 is 2 marks)
14
A different hockey team's manager recorded the number of goals scored in each match of a season. The mean number of goals scored per match was 1.5. One frequency value, t, is unknown. Find the value of t.
(Total for Question 14 is 4 marks)
15
A group of students recorded the time, t minutes, each spent on homework one evening. The estimated mean time was 25 minutes. One frequency value, p, is unknown. Find the value of p.
(Total for Question 15 is 5 marks)
16
A cafe is choosing between two vending machines. Each machine was tested with 30 customers, and the waiting time for a drink, w seconds, was recorded for each customer. Calculate an estimate of the mean waiting time for each machine, and use your answers to advise the cafe which machine has the shorter typical waiting time. You must show your working.
(Total for Question 16 is 4 marks)
17
The times, in seconds, taken by 20 runners to finish a sprint were recorded.
(a)Use the data to complete a grouped frequency table with the class intervals 9 ≤ t < 11, 11 ≤ t < 13 and 13 ≤ t < 15.(2)
(b)Calculate an estimate of the mean sprint time. Give your answer correct to 2 decimal places.(4)
(Total for Question 17 is 6 marks)
Mark scheme · 4.18 Averages from Frequency Tables
Question 1
(a) B1 1 cao
(a) Answer: 1
(b) B1 4 cao
(b) Answer: 4
Question 2
M1 correct cumulative frequencies used to locate the 15th and 16th values (6, 16, 24, 28, 30) oe
A1 1 cao
Answer: 1
Question 3
M1 at least 4 correct values of f x x, or sight of sum(fx) = 46
M1 46 / 30 (dependent on previous M1)
A1 1.53 awrt oe
Answer: 1.53 (exact value 23/15)
Question 4
(a) B1 1 cao
(a) Answer: 1
(b) M1 correct cumulative frequencies used to locate the 10th and 11th values (2, 9, 13, 18, 20) oe
(b) A1 2 cao
(b) Answer: 2
(c) M1 at least 4 correct values of f x x, or sight of sum(fx) = 38
(c) M1 38 / 20 (dependent on previous M1)
(c) A1 1.9 cao
(c) Answer: 1.9
Question 5
M1 at least 5 correct values of f x x, or sight of sum(fx) = 63
M1 63 / 25 (dependent on previous M1)
A1 2.52 cao
Answer: 2.52
Question 6
(a) B1 30 ≤ t < 40 cao
(a) Answer: 30 ≤ t < 40
(b) B1 21 cao
(b) Answer: 21
(c) M1 correct cumulative frequencies used to locate the 20th and 21st values (6, 21, 32, 40) oe
(c) A1 30 ≤ t < 40 cao
(c) Answer: 30 ≤ t < 40
Question 7
M1 midpoints 25, 35, 45, 55 identified
M1 sum(f x midpoint) = 1610 oe
M1 1610 / 40 (dependent on both previous M marks)
A1 40.25 awrt oe
Answer: 40.25 seconds
Question 8
(a) M1 midpoints 1, 3, 5, 7, 9 identified
(a) M1 sum(f x midpoint) = 172 oe
(a) M1 172 / 50 (dependent on both previous M marks)
(a) A1 3.44 awrt oe
(a) Answer: 3.44 km
(b) B1 correct reason, e.g. the exact distance travelled by each person within a class interval is not known, only the interval it lies in, so the midpoint (an assumed value) is used to represent every person in that class, oe
(b) Answer: The exact individual distances are not known, only which class interval each one falls in, so the midpoint of each class is used as an assumed value instead of the real data.
Question 9
M1 50 - (8 + 15 + 9 + 3) oe
A1 15 cao
Answer: y = 15
Question 10
M1 correct use of negative values, sum(fx) = -19 oe
M1 -19 / 20 (dependent on previous M1)
A1 -0.95 cao
Answer: -0.95 degrees C
Question 11
M1 at least 4 correct values of f x x, or sight of sum(fx) = 117
M1 117 / 20 (dependent on previous M1)
A1 5.85 cao
Answer: 5.85
Question 12
(a) M1 at least 4 correct values of f x x, or sight of sum(fx) = 137
(a) M1 137 / 20 (dependent on previous M1)
(a) A1 6.85 cao
(a) Answer: 6.85
(b) B1 Class A range = 4 cao
(b) B1 Class B range = 3 cao
(b) Answer: Class A range = 4; Class B range = 3
Question 13
B1 correct comparison of the means with correct direction, e.g. Class B has a higher mean (6.85 > 5.85), so Class B scored higher on average, oe
B1 correct comparison of the ranges with correct direction, e.g. Class B has a smaller range (3 < 4), so Class B's scores were more consistent, oe
Answer: Class B did better on average, since its mean (6.85) is higher than Class A's mean (5.85). Class B's scores were also more consistent, since its range (3) is smaller than Class A's range (4).
Question 14
M1 sum(fx) = 15 + 2t oe
M1 sum(f) = 14 + t oe
M1 correct equation formed and rearranged, e.g. (15 + 2t) = 1.5(14 + t) oe
A1 t = 12 cao
Answer: t = 12
Question 15
M1 midpoints 7.5, 22.5, 37.5, 52.5 identified
M1 sum(f x midpoint) = 495 + 22.5p oe
M1 total frequency = 16 + p oe
M1 correct equation formed and rearranged, e.g. 495 + 22.5p = 25(16 + p) oe
A1 p = 38 cao
Answer: p = 38
Question 16
M1 estimate for Machine A: sum(f x midpoint) = 410, mean = 410 / 30 = 13.67 awrt oe
M1 estimate for Machine B: sum(f x midpoint) = 490, mean = 490 / 30 = 16.33 awrt oe
A1 both means correct, awrt 13.67 and awrt 16.33
B1 correct conclusion, Machine A, with valid reason based on the two calculated means (ft their means)
Answer: Machine A: 13.67 seconds; Machine B: 16.33 seconds; recommend Machine A, as it has the shorter mean waiting time.
Question 17
(a) B1 at least two of the three frequencies correct
(a) B1 all three frequencies correct: 6, 6, 8
(a) Answer: 9 ≤ t < 11: 6; 11 ≤ t < 13: 6; 13 ≤ t < 15: 8
(b) M1 midpoints 10, 12, 14 identified (ft their frequencies from (a))
(b) M1 sum(f x midpoint) = 244 oe (ft their frequencies)
(b) M1 244 / 20 (dependent on both previous M marks)