Statistical sampling and data presentationAQA A-Level Maths: Topic test

20 questions, 54 marks

AQA A-Level Maths

Statistical sampling and data presentation topic test

Total 54 marks

Name

Class

Date

1
A water company wants to estimate the mean daily water use of the 800 households in a village. Each household has a customer number from 1 to 800. An officer uses a random number generator to produce 40 different numbers between 1 and 800 and surveys the households with those numbers.
(a)
Which of the following is the population in this investigation?
[1 mark]
AThe 40 households that are surveyed
BAll 800 households in the village
CThe water use of one household
DThe mean daily water use of the sample
(b)
Which sampling technique is the officer using?
[1 mark]
ASimple random sampling
BOpportunity sampling
CA census
DSampling only the first 40 customer numbers
(c)
Give two advantages of this method over opportunity sampling.
[2 marks]
Total for question 1: 4 marks
2
The masses, $m$ grams, of 70 apples are summarised in a histogram. The classes and frequency densities are: $100\le m<120$ with frequency density $0.5$ , $120\le m<140$ with frequency density $1.5$ , $140\le m<160$ with frequency density $1.0$ and $160\le m<200$ with frequency density $0.25$ .
(a)
How many apples have a mass in the class $120\le m<140$ ?
[1 mark]
A $1.5$
B $20$
C $30$
D $60$
(b)
Assuming the masses are evenly spread within each class, how many apples have a mass below $130$ g?
[1 mark]
A $20$
B $35$
C $40$
D $25$
(c)
Estimate the number of apples with a mass between $150$ g and $180$ g.
[2 marks]
Total for question 2: 4 marks
3
A café records the number of customers, $x$ , who arrive in each of 20 half-hour periods. The summary statistics are $\sum x=240$ and $\sum x^2=3180$ .
(a)
Calculate the mean and standard deviation of the number of customers per half-hour period.
[3 marks]
(b)
Two further half-hour periods are recorded, with $20$ customers and $4$ customers. Find the mean and standard deviation of all 22 periods.
[4 marks]
Total for question 3: 7 marks
4
A manufacturer measures the battery life, in hours, of 12 randomly chosen tablets: 8.1, 8.4, 8.6, 8.8, 9.0, 9.1, 9.3, 9.4, 9.6, 9.8, 10.1 and 14.6. Here a value is treated as an outlier if it is more than $1.5\times$ the interquartile range above the upper quartile or below the lower quartile.
(a)
(i) Find the median and the interquartile range. (ii) Show that the value $14.6$ is an outlier. (iii) Give one possible reason for the value $14.6$ and state what the manufacturer should do before deciding whether to remove it.
[6 marks]
(b)
The value $14.6$ is removed after checking. (i) Find the mean of all 12 values and of the remaining 11 values. (ii) Find the median of the remaining 11 values. (iii) Using your answers, state which of the mean and median is less affected by the outlier, and which you would quote as the typical battery life.
[6 marks]
Total for question 4: 12 marks
5
A researcher records, for 15 towns, the number of pharmacies in each town and the number of reported cases of flu in a winter. The points on a scatter diagram show positive correlation. The towns have between 3 and 40 pharmacies.
(a)
Which statement is best supported by this information?
[1 mark]
ATowns with more pharmacies tend to have more reported cases of flu
BOpening a pharmacy in a town causes more flu cases there
CEvery town with more pharmacies has more flu cases than every town with fewer
DThere is no relationship between pharmacies and flu cases
(b)
A regression line of cases on pharmacies is drawn through the points. For which number of pharmacies would a prediction from the line be most reliable?
[1 mark]
A $0$
B $100$
C $25$
D $300$
(c)
Explain why the researcher cannot conclude that opening more pharmacies in a town leads to more flu cases, and suggest one other variable that could explain the correlation.
[2 marks]
Total for question 5: 4 marks
6
A researcher wants to estimate how long people in a city wait for a bus. She stands at the central bus station on a Monday morning and asks the first 50 people who arrive how long they waited.
(a)
Which sampling technique is she using?
[1 mark]
ASimple random sampling
BA census
CTaking every fifth person
DOpportunity sampling
(b)
Which is the main weakness of this sample?
[1 mark]
AThe sample is too large
BIt may not represent all bus users, for example at other times and places
CRandom numbers were not used to choose the day
DThe people asked will all have waited the same time
(c)
A colleague repeats the survey with a different 50 people at the same station on the following Monday. Explain why the two samples may lead to different conclusions about waiting times in the city.
[2 marks]
Total for question 6: 4 marks
7
A student takes a random sample of 9 days from the large data set for one UK station in August and records the daily total sunshine, in hours: 4.2, 8.9, 0.6, 11.3, 7.5, 9.8, 2.1, 10.4 and 6.7.
(a)
Use your calculator to find the mean and the standard deviation (dividing by $n$ ) of these values, and state why the mean of the sample may differ from the mean for all days in August.
[3 marks]
(b)
The student wants to claim that August 2015 was sunnier than August 1987 at this station using the large data set. State three checks or steps she should take so that the comparison is fair, and one reason why her conclusion would still be an inference rather than a certainty.
[4 marks]
Total for question 7: 7 marks
8
A property website records, for 40 houses in a town, the floor area $a$ m $^2$ and the asking price $p$ in thousands of pounds. A scatter diagram shows positive correlation overall, but the points form two separate groups: 22 terraced houses with areas from 60 to 110 m $^2$ and 18 detached houses with areas from 140 to 260 m $^2$ . A regression line fitted to all 40 points is $p=1.8a+25$ .
(a)
(i) Interpret the value $1.8$ in context. (ii) Comment on the value $25$ . (iii) Use the line to estimate the asking price of a house with floor area $120$ m $^2$ and comment on the reliability of the estimate.
[6 marks]
(b)
The 22 terraced houses have mean asking price £185 000 and standard deviation £20 000. The 18 detached houses have mean asking price £420 000 and standard deviation £55 000. (i) Find the mean asking price of all 40 houses. (ii) Explain why this mean is not a good description of a typical house in the town. (iii) Compare the variation in asking prices for the two types of house, in absolute terms and relative to their means, and say what the website should report.
[6 marks]
Total for question 8: 12 marks

End of questions

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).