Statistics: Data presentation and interpretationEdexcel A-Level Maths: Topic test

1. 1
   The lengths, in minutes, of $120$ films are summarised in a histogram. The frequencies are: $60\leqslant l<90$, $18$ films; $90\leqslant l<100$, $30$ films; $100\leqslant l<120$, $48$ films; $120\leqslant l<180$, $24$ films.

   (a)

   What is the frequency density for the class $120\leqslant l<180$?

   [1 mark]

   • A$24$
   • B$0.4$
   • C$2.5$
   • D$4$

   (b)

   On the histogram the bar for $100\leqslant l<120$ is drawn $4.8$ cm tall. How tall should the bar for $60\leqslant l<90$ be drawn?

   [1 mark]

   • A$0.6$ cm
   • B$3.6$ cm
   • C$1.8$ cm
   • D$1.2$ cm

   (c)

   Estimate the number of films with a length of at least $95$ minutes but less than $110$ minutes.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The times, $x$ seconds, taken by $12$ sprinters to run $100$ m are summarised by $\sum x=150$ and $\sum x^2=1890$.

   (a)

   What is the mean time?

   [1 mark]

   • A$12.5$ s
   • B$150$ s
   • C$12.0$ s
   • D$157.5$ s

   (b)

   What is the standard deviation of the times, to $3$ significant figures?

   [1 mark]

   • A$1.25$ s
   • B$15.0$ s
   • C$1.12$ s
   • D$12.6$ s

   (c)

   Each time is converted to a score using $y=4x-10$. Find the mean and the standard deviation of the scores $y$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The distances driven in one day by $60$ taxi drivers are summarised as follows: $0\leqslant d<50$, $6$ drivers; $50\leqslant d<100$, $15$ drivers; $100\leqslant d<150$, $21$ drivers; $150\leqslant d<200$, $12$ drivers; $200\leqslant d<300$, $6$ drivers. Distances are in km.

   (a)

   Use linear interpolation to estimate the median distance, to $3$ significant figures.

   [3 marks]

   (b)

   Estimate the mean and the standard deviation of the distances.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An ice-cream kiosk records its sales on $30$ summer days. The daily sales, $y$ hundred pounds, have minimum $1.9$, lower quartile $4.2$, median $5.8$, upper quartile $7.0$ and maximum $12.6$. The kiosk owner also records the maximum temperature, $x$ $^\circ$C, on each day. The values of $x$ range from $18$ to $32$, and the regression line of $y$ on $x$ is $y=-4.8+0.35x$. A value is classed as an outlier if it is more than $1.5\times$ the interquartile range above the upper quartile or below the lower quartile.

   (a)

   Show that the maximum daily sales figure is an outlier but the minimum is not. Describe how the sales data would be shown on a box plot, and what the owner should do before deciding whether to remove the outlier.

   [6 marks]

   (b)

   State which is the explanatory variable and interpret the gradient of the regression line in context. Use the line to predict the sales when the temperature is $28$ $^\circ$C, and explain why a prediction for $40$ $^\circ$C would be unreliable.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A box and whisker plot shows the daily number of steps, in thousands, taken by $40$ people. The minimum is $2$, the lower quartile is $5$, the median is $8$, the upper quartile is $11$ and the maximum is $21$. A value is an outlier if it is more than $1.5\times$ the interquartile range above the upper quartile or below the lower quartile.

   (a)

   What is the interquartile range?

   [1 mark]

   • A$3$
   • B$9$
   • C$19$
   • D$6$

   (b)

   Which statement is correct?

   [1 mark]

   • AThe maximum value is an outlier, because it is greater than $11+1.5\times6=20$.
   • BThe maximum value is not an outlier, because it is less than $11+3\times6=29$.
   • CThe minimum value is an outlier, because it is less than $5-1.5\times6=-4$.
   • DThere are no outliers, because the maximum is less than twice the upper quartile ($21<22$).

   (c)

   Describe the skewness of the data, using the quartiles, the median and the extreme values to justify your answer.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The stopping distance, $s$ metres, of a car travelling at $v$ m s$^{-1}$ is modelled by $s=kv^n$. When $\log_{10}s$ is plotted against $\log_{10}v$ the points lie close to a straight line with gradient $2$ and vertical intercept $-1.6$.

   (a)

   What is the value of $n$?

   [1 mark]

   • A$0.5$
   • B$-1.6$
   • C$2$
   • D$100$

   (b)

   What is the value of $k$, to $2$ significant figures?

   [1 mark]

   • A$-1.6$
   • B$0.025$
   • C$1.6$
   • D$40$

   (c)

   Use the model to estimate the stopping distance at $30$ m s$^{-1}$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A weather station records the daily mean temperature for $365$ days. The mean is $12.4$ $^\circ$C and the standard deviation is $4.1$ $^\circ$C. A reading is classed as an outlier if it is more than $3$ standard deviations from the mean. Three readings look unusual: $-3.0$ $^\circ$C, $26.0$ $^\circ$C and $37.5$ $^\circ$C.

   (a)

   Determine which of the three readings are outliers.

   [3 marks]

   (b)

   The reading of $37.5$ $^\circ$C is found to be caused by a faulty sensor and is removed. Find the new mean of the remaining readings, to $4$ significant figures, and explain how the other two outliers should be treated.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The marks, out of $100$, of $40$ students in a mock test are summarised as follows: $0\leqslant m<20$, $4$ students; $20\leqslant m<30$, $8$ students; $30\leqslant m<40$, $14$ students; $40\leqslant m<60$, $10$ students; $60\leqslant m<100$, $4$ students.

   (a)

   Find the frequency density of each class, and use linear interpolation to estimate the median and the interquartile range.

   [6 marks]

   (b)

   Estimate the mean and standard deviation of the marks. Hence comment on the skewness, and decide whether a mark of $95$ is an outlier if outliers are values more than $3$ standard deviations above the mean.

   [6 marks]

   Total for question 8: 12 marks

End of questions