Measures of dispersionEdexcel International A Level Maths: Subtopic test

1. 1
   The reaction times, in tenths of a second, of eight athletes were 2, 4, 4, 4, 5, 5, 7 and 9.

   (a)

   Find the variance of the eight values.

   [1 mark]

   • A$2$
   • B$4$
   • C$29$
   • D$4.57$

   (b)

   Each value is multiplied by 3 and then 2 is added. Find the standard deviation of the new values.

   [1 mark]

   • A$8$
   • B$12$
   • C$36$
   • D$6$

   (c)

   Find the interquartile range of the eight values.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sample of 20 delivery times, $x$ minutes, has $\sum x=140$ and $\sum x^2=1160$. A second sample of delivery times, from another firm, has the same mean but a standard deviation of 5 minutes.

   (a)

   Find the variance of the first sample.

   [1 mark]

   • A$9$
   • B$51$
   • C$58$
   • D$3$

   (b)

   Which statement compares the two samples correctly?

   [1 mark]

   • AThe second firm's times are more consistent because their standard deviation is larger.
   • BThe second firm's times have a higher mean because their standard deviation is larger.
   • CThe second firm's times are more spread out about the mean, so they are less consistent.
   • DThe two firms are equally consistent because their means are equal.

   (c)

   A 21st delivery time of 7 minutes is added to the first sample. Find the new standard deviation.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The waiting times, in minutes, of 80 patients at a clinic are summarised as follows: under 10 minutes, 6 patients; 10 to under 20 minutes, 24 patients; 20 to under 30 minutes, 28 patients; 30 to under 40 minutes, 16 patients; 40 to under 60 minutes, 6 patients. Assume that times are spread evenly within each class.

   (a)

   Estimate the interquartile range of the waiting times.

   [3 marks]

   (b)

   Estimate the 10th to 90th interpercentile range.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two machines fill bottles with a target volume of 500 ml. For machine A, the volumes $x$ ml of 12 bottles are coded using $y=x-500$, giving $\sum y=18$ and $\sum y^2=70$. For machine B, a sample of 12 bottles has mean volume 500.6 ml and standard deviation 0.8 ml.

   (a)

   (i) Find the mean and the standard deviation of the volumes for machine A.
   (ii) Compare the performance of the two machines.

   [6 marks]

   (b)

   A further bottle from machine A is found to contain 509 ml. Find the new mean and standard deviation for the 13 bottles from machine A, and describe the effect of this bottle on each.

   [6 marks]

   Total for question 4: 12 marks

End of questions