The large data setAQA A-Level Maths: Subtopic test

1. 1
   A large data set records, for each new car sold in a year, the fuel type (petrol, diesel or hybrid), the number of doors, the engine size in litres and the CO₂ emissions in g/km. Some records have the CO₂ emissions blank.

   (a)

   Which of the four variables is a continuous quantitative variable?

   [1 mark]

   • AFuel type
   • BEngine size
   • CNumber of doors
   • DCO₂ emissions recorded as 'low', 'medium' or 'high'

   (b)

   Before finding the mean CO₂ emissions, a student replaces each blank by 0. Compared with the mean of the recorded values, the new mean will be

   [1 mark]

   • Atoo high, because there are more values
   • Bunchanged, because 0 adds nothing to the total
   • Ctoo low, because the zeros are not genuine readings
   • Dtoo low only if the data are skewed

   (c)

   Name a suitable diagram for displaying the fuel type of the cars, and explain why a histogram would not be suitable.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A student takes a random sample of 11 petrol cars from a large data set and records the CO₂ emissions in g/km: 110, 114, 118, 121, 124, 126, 129, 131, 134, 138, 205. Quartiles are the medians of the lower and upper halves of the ordered data, leaving out the median. A value is an outlier if it is more than $1.5\times$ IQR above the upper quartile or below the lower quartile.

   (a)

   Find the median CO₂ emissions of the sample, in g/km.

   [1 mark]

   • A126
   • B131.8
   • C124
   • D129

   (b)

   Find the interquartile range of the sample, in g/km.

   [1 mark]

   • A24
   • B13
   • C95
   • D16

   (c)

   Show that 205 g/km is an outlier.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student uses a large data set of daily weather records from one station. Missing readings are shown by the value $-99$. A random sample of 40 valid days gives $\sum x=612$ and $\sum x^2=9774$ for the daily mean temperature $x$ in °C. A value is an outlier if it lies more than 2 standard deviations from the mean.

   (a)

   Explain what would go wrong if the readings of $-99$ were included when calculating the mean and standard deviation, and what the student should do.

   [3 marks]

   (b)

   Use the sample to find the mean and standard deviation of the temperature, and hence determine whether a day with mean temperature $25.5$°C is an outlier.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A student compares daily mean wind speeds, in knots, at two locations, A and B, using random samples of 30 days from each in a large data set. Location A: mean 9.4, standard deviation 2.1, median 9.2, interquartile range 2.8. Location B: mean 11.2, standard deviation 4.8, median 8.6, lower quartile 6.9, upper quartile 10.0.

   (a)

   Compare the wind speeds at the two locations, referring to a typical day and to variability, and comment on what the summary statistics suggest about the shape of the data at B.

   [6 marks]

   (b)

   A day at location B has a mean wind speed of $20.5$ knots. Test whether it is an outlier using (i) the rule 'more than $1.5\times$ IQR above the upper quartile' and (ii) the rule 'more than 2 standard deviations above the mean'. Comment on your results.

   [6 marks]

   Total for question 4: 12 marks

End of questions