A policeman records the speed of the traffic on a busy road with a 30 mph speed limit. He records the speeds of a sample of 450 cars. The histogram in Figure 2 represents the results. (a) Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the sample. (b) Estimate the value of the mean speed of the cars in the sample. (c) Estimate, to 1 decimal place, the value of the median speed of the cars in the sample. (d) Comment on the shape of the distribution. Give a reason for your answer. (e) State, with a reason, whether the estimate of the mean or the median is a better representation of the average speed of the traffic on the road.

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A policeman records the speed of the traffic on a busy road with a 30 mph speed limit - Edexcel - A-Level Maths Statistics - Question 5 - 2012 - Paper 2

Question 5

A policeman records the speed of the traffic on a busy road with a 30 mph speed limit. He records the speeds of a sample of 450 cars. The histogram in Figure 2 repre... show full transcript

Worked Solution & Example Answer:A policeman records the speed of the traffic on a busy road with a 30 mph speed limit - Edexcel - A-Level Maths Statistics - Question 5 - 2012 - Paper 2

Step 1

Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the sample.

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Answer

To find the number of cars exceeding the speed limit of 30 mph, we need to consider speeds of 35 mph and above. From the histogram:

For the interval 30-35 mph, the frequency density is approximately 0.25. The corresponding width is 5 mph, contributing about: $0.25 \times 5 = 1.25$ Therefore, the estimated number of cars in this interval is $1.25 \times 450 = 112.5$ .
For the interval 35-40 mph, the frequency density is approximately 0.6. Using the same method: $0.6 \times 5 = 3$ , which gives $3 \times 450 = 1350$ .
For the interval 40-45 mph, the frequency density gives $1 \times 5 = 5$ , contributing to $5 \times 450 = 2250$ .

Summing these values, the total number of cars exceeding the speed limit by at least 5 mph is: $112.5 + 135 + 225 = 472.5$ . Thus, approximately 473 cars were exceeding the speed limit.

Step 2

Estimate the value of the mean speed of the cars in the sample.

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Answer

To estimate the mean speed, we calculate the mid-point of each interval, multiply by the frequency density and then find the cumulative frequency:

For 0-5 mph: Mid-point = 2.5, Contribution = $2.5 \times (0 \times 5) = 0$ .
For 5-10 mph: Mid-point = 7.5, Contribution = $7.5 \times (0.05 \times 5) = 0.1875$ .
For 10-15 mph: Mid-point = 12.5, Contribution = $12.5 \times 1 = 12.5$ .
For 15-20 mph: Mid-point = 17.5, Contribution = $17.5 \times 2 = 35$ .
For 20-25 mph: Mid-point = 22.5, Contribution = $22.5 \times 3 = 67.5$ .
For 25-30 mph: Mid-point = 27.5, Contribution = $27.5 \times 3.5 = 96.25$ .
For 30-35 mph: Mid-point = 32.5, Contribution = $32.5 \times (1 \times 5) = 162.5$ .

Total Mean Estimate: $\frac{0 + 0 + 12.5 + 35 + 67.5 + 96.25 + 162.5}{450} \approx 28.8$ mph.

Step 3

Estimate, to 1 decimal place, the value of the median speed of the cars in the sample.

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Answer

To find the median, we need the total frequency which is 450, i.e., the median is in the 225th position.

Cumulative frequency table based on intervals:

0-5 mph: 0
5-10 mph: 5
10-15 mph: 5
15-20 mph: 7
20-25 mph: 10
25-30 mph: 12
30-35 mph: 15

Contributing to the median, we find that the median speed lies in the interval 25-30 mph. Thus, the estimated median speed is approximately $28.1$ .

Step 4

Comment on the shape of the distribution. Give a reason for your answer.

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Answer

The shape of the distribution is positively skewed, as indicated by the longer tail on the right side. This suggests that there are many cars traveling at lower speeds (below the mean) with few cars traveling at higher speeds. In a symmetrical distribution, both ends would mirror each other; here, the mean is greater than the median, reinforcing that the distribution is not symmetrical.

Step 5

State, with a reason, whether the estimate of the mean or the median is a better representation of the average speed of the traffic on the road.

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Answer

The median is a better estimate of the average speed because it is less affected by extreme values (outliers), whereas the mean can be skewed by particularly high speeds, given the right-skewed nature of the distribution. Therefore, the median provides a more reliable measure of central tendency in this scenario.

A policeman records the speed of the traffic on a busy road with a 30 mph speed limit - Edexcel - A-Level Maths Statistics - Question 5 - 2012 - Paper 2

Worked Solution & Example Answer:A policeman records the speed of the traffic on a busy road with a 30 mph speed limit - Edexcel - A-Level Maths Statistics - Question 5 - 2012 - Paper 2

Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the sample.

Estimate the value of the mean speed of the cars in the sample.

Estimate, to 1 decimal place, the value of the median speed of the cars in the sample.

Comment on the shape of the distribution. Give a reason for your answer.

State, with a reason, whether the estimate of the mean or the median is a better representation of the average speed of the traffic on the road.

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