Write down the midpoint of the $60 < s \leqslant 70$ interval.

Use your graphic display calculator to find an estimate for

(i)     the mean speed of the vehicles;

(ii)     the standard deviation of the speeds of the vehicles.

Write down the number of vehicles whose speed is less than or equal to ${\text{60 km }}{{\text{h}}^{ - 1}}$.

Consider the cumulative frequency table below.

Write down the value of $a$ , of $b$ and of $c$ .

Consider the cumulative frequency table below.

Draw a cumulative frequency graph for the information from the table. Use $1$ cm to represent ${\text{10 km }}{{\text{h}}^{ - 1}}$ on the horizontal axis and $1$ cm to represent $10$ vehicles on the vertical axis.

Use your cumulative frequency graph to estimate

(i)     the median speed of the vehicles;

(ii)     the number of vehicles that are travelling at a speed less than or equal to ${\text{65 km }}{{\text{h}}^{ - 1}}$.

All drivers whose vehicle’s speed is greater than one standard deviation above the speed limit of ${\text{50 km }}{{\text{h}}^{ - 1}}$ will be fined.

Use your graph to estimate the number of drivers who will be fined.

$65$     (A1)

[1 mark]

(i)      $54{\text{ (km }}{{\text{h}}^{ - 1}})$     (G2)

Note: If the answer to part (b)(i) is consistent with the answer to part (a) then award (G2)(ft) even if no working seen.

(ii)     $19.2$ ($19.2093 \ldots$)     (G1)

Note: Accept $19$, do not accept $20$.

[3 marks]

$76$     (A1)

[1 mark]

$a = 76$, $b = 98$     (A1)(ft)

Note: Follow through from their answer to part (c) for $a$ and $b =$ their $a + 22$ .

$c = 118$     (A1)

[2 marks]

     (A1)(A1)(ft)(A1)(ft)(A1)

Notes: Award (A1) for axes labelled and correct scales. If the axes are reversed do not award this mark but follow through. Award (A2)(ft) for their 6 points correct, (A1)(ft) for at least 3 of these points correct. Award (A1) for smooth curve drawn through all points including ($0$, $0$). If either the $x$ or the $y$ axis has a break in it to zero, do not award this final mark.

[4 marks]

(i)     $57$ ${\text{(km }}{{\text{h}}^{ - 1}})$ $( \pm 2)$     (M1)(A1)(ft)(G2)

Note: Award (M1) for clear indication of median on their graph. Follow through from their graph. If their answer is consistent with their incorrect graph but there is no working present on graph then no marks are awarded.

(ii)     $90$ vehicles $( \pm 2)$     (M1)(A1)(ft)(G2)

Note: Award (M1) for clear indication of method on their graph. Follow through from their graph. If their answer is consistent with their incorrect graph but there is no working present on graph then no marks are awarded.

[4 marks]

$50 + 19.2 = 69.2$     (A1)(ft)

$24$ $( \pm 2)$ drivers will be fined     (M1)(A1)(ft)(G2)

Notes: Follow through from their graph and from their part (b)(ii). Award (M1) for indication of method on their graph. If their answer is consistent with their incorrect graph but there is no working present on graph then no marks are awarded.

[3 marks]

The great majority of candidates found this question to be a good start to the paper. The different class widths seemed to cause more problems to the teachers commenting on the G2 forms than to the students. However, 1(b) was a discriminator at the grade 4 level. Most candidates were successful in drawing the cumulative frequency curve or attempting to do so. There were a small number who clearly had never had any experience with this type of graph. A common error was the incorrect plotting of points at the interval midpoints. Weaker candidates plotted bar charts.

A small number of candidates did not use the graph paper provided, preferring instead to use lined paper. This is to be strongly discouraged since no judgment will be made about the scale used or the accuracy of the plotted points. Similarly, the graph will not be used to benefit students whose answers lie outside accepted tolerances but who have shown working. Drawing an accurate graph requires the use of graph paper.

The great majority of candidates found this question to be a good start to the paper. The different class widths seemed to cause more problems to the teachers commenting on the G2 forms than to the students. However, 1(b) was a discriminator at the grade 4 level. Most candidates were successful in drawing the cumulative frequency curve or attempting to do so. There were a small number who clearly had never had any experience with this type of graph. A common error was the incorrect plotting of points at the interval midpoints. Weaker candidates plotted bar charts.

A small number of candidates did not use the graph paper provided, preferring instead to use lined paper. This is to be strongly discouraged since no judgment will be made about the scale used or the accuracy of the plotted points. Similarly, the graph will not be used to benefit students whose answers lie outside accepted tolerances but who have shown working. Drawing an accurate graph requires the use of graph paper.

The great majority of candidates found this question to be a good start to the paper. The different class widths seemed to cause more problems to the teachers commenting on the G2 forms than to the students. However, 1(b) was a discriminator at the grade 4 level. Most candidates were successful in drawing the cumulative frequency curve or attempting to do so. There were a small number who clearly had never had any experience with this type of graph. A common error was the incorrect plotting of points at the interval midpoints. Weaker candidates plotted bar charts.

A small number of candidates did not use the graph paper provided, preferring instead to use lined paper. This is to be strongly discouraged since no judgment will be made about the scale used or the accuracy of the plotted points. Similarly, the graph will not be used to benefit students whose answers lie outside accepted tolerances but who have shown working. Drawing an accurate graph requires the use of graph paper.

The great majority of candidates found this question to be a good start to the paper. The different class widths seemed to cause more problems to the teachers commenting on the G2 forms than to the students. However, 1(b) was a discriminator at the grade 4 level. Most candidates were successful in drawing the cumulative frequency curve or attempting to do so. There were a small number who clearly had never had any experience with this type of graph. A common error was the incorrect plotting of points at the interval midpoints. Weaker candidates plotted bar charts.

A small number of candidates did not use the graph paper provided, preferring instead to use lined paper. This is to be strongly discouraged since no judgment will be made about the scale used or the accuracy of the plotted points. Similarly, the graph will not be used to benefit students whose answers lie outside accepted tolerances but who have shown working. Drawing an accurate graph requires the use of graph paper.

The great majority of candidates found this question to be a good start to the paper. The different class widths seemed to cause more problems to the teachers commenting on the G2 forms than to the students. However, 1(b) was a discriminator at the grade 4 level. Most candidates were successful in drawing the cumulative frequency curve or attempting to do so. There were a small number who clearly had never had any experience with this type of graph. A common error was the incorrect plotting of points at the interval midpoints. Weaker candidates plotted bar charts.

A small number of candidates did not use the graph paper provided, preferring instead to use lined paper. This is to be strongly discouraged since no judgment will be made about the scale used or the accuracy of the plotted points. Similarly, the graph will not be used to benefit students whose answers lie outside accepted tolerances but who have shown working. Drawing an accurate graph requires the use of graph paper.

The great majority of candidates found this question to be a good start to the paper. The different class widths seemed to cause more problems to the teachers commenting on the G2 forms than to the students. However, 1(b) was a discriminator at the grade 4 level. Most candidates were successful in drawing the cumulative frequency curve or attempting to do so. There were a small number who clearly had never had any experience with this type of graph. A common error was the incorrect plotting of points at the interval midpoints. Weaker candidates plotted bar charts.

A small number of candidates did not use the graph paper provided, preferring instead to use lined paper. This is to be strongly discouraged since no judgment will be made about the scale used or the accuracy of the plotted points. Similarly, the graph will not be used to benefit students whose answers lie outside accepted tolerances but who have shown working. Drawing an accurate graph requires the use of graph paper.

The great majority of candidates found this question to be a good start to the paper. The different class widths seemed to cause more problems to the teachers commenting on the G2 forms than to the students. However, 1(b) was a discriminator at the grade 4 level. Most candidates were successful in drawing the cumulative frequency curve or attempting to do so. There were a small number who clearly had never had any experience with this type of graph. A common error was the incorrect plotting of points at the interval midpoints. Weaker candidates plotted bar charts.

A small number of candidates did not use the graph paper provided, preferring instead to use lined paper. This is to be strongly discouraged since no judgment will be made about the scale used or the accuracy of the plotted points. Similarly, the graph will not be used to benefit students whose answers lie outside accepted tolerances but who have shown working. Drawing an accurate graph requires the use of graph paper.