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In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month - Edexcel - A-Level Maths Statistics - Question 5 - 2009 - Paper 1
Question 5
In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month. The results are summaris... show full transcript
Worked Solution & Example Answer:In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month - Edexcel - A-Level Maths Statistics - Question 5 - 2009 - Paper 1
Step 1
Calculate the width and height of the rectangle representing the group (16 - 25) hours.
Answer
To find the width and height, we examine the group (16 - 25) hours. The group spans from 16 to 25 hours which gives a width of:
Width = 25 - 16 = 9 hours.
For height, since the frequency of teenagers in the group (16 - 25) hours is 15, we calculate the height of each bar in the histogram using the scaling factor based on the width given for the (8 - 10) hours, which is 3 cm high and 1.5 cm wide. This gives:
Height = (Frequency / Total Frequency) * (Width of reference rectangle / Width of group) Height = (15 / 104) * (3 / 1.5) = 0.5 cm.
Step 2
Use linear interpolation to estimate the median and interquartile range.
Answer
For the cumulative frequency of 104:
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The median is the 52nd value. Since the group (16 - 25) contains the median, we find:
- Median (Q2) lies between 16 and 25. Estimating using the cumulative frequency data:
- The cumulative frequency before 16 is 54 (20 + 16 + 18). Thus, interpolation gives: Q2 = 16 + (52 - 54)/(15) * (25 - 16) = 20.3.
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For the first quartile (Q1), locate the 26th value: Q1 lies between 11 and 15, interpolate: Q1 = 11 + (26 - 18)/(25) * (15 - 11) = 13.6
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For the third quartile (Q3), locate the 78th value: Q3 lies between 16 and 25, interpolate: Q3 = 16 + (78 - 54)/(15) * (25 - 16) = 21.4
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The interquartile range (IQR) is: IQR = Q3 - Q1 = 21.4 - 13.6 = 7.8.
Step 3
Estimate the mean and standard deviation of the number of hours spent shopping.
Answer
The mean can be calculated as follows:
- Calculate the mid-point of each group multiplied by its frequency:
Mean = (Σ(mid-point × frequency) / total frequency) Using the data: = (2.7520 + 6.516 + 918 + 1325 + 20.515 + 3810) / 104 = 12.8.
- To find the standard deviation: Variance = (Σ(frequency × (mid-point - mean)²) / total frequency). Calculate and find: Standard deviation = sqrt(Variance) ≈ 9.88.
Step 4
Step 5
State, giving a reason, which average and measure of dispersion you would recommend to use to summarise these data.
Answer
I would recommend using the median as the average and the interquartile range (IQR) as the measure of dispersion. This is suitable as the data is skewed, minimizing the effects of outliers.