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Summarised below are the distances, to the nearest mile, travelled to work by a random sample of 120 commuters - Edexcel - A-Level Maths Statistics - Question 4 - 2007 - Paper 1
Question 4
Summarised below are the distances, to the nearest mile, travelled to work by a random sample of 120 commuters. Distance (to the nearest mile) Number of commu... show full transcript
Worked Solution & Example Answer:Summarised below are the distances, to the nearest mile, travelled to work by a random sample of 120 commuters - Edexcel - A-Level Maths Statistics - Question 4 - 2007 - Paper 1
Step 1
describe its shape
Answer
The shape of the distribution is positively skewed. This is indicated by a longer tail on the right side, as there are more commuters traveling shorter distances compared to those traveling longer distances. The majority of data points cluster at the lower end of the distance range.
Step 2
use linear interpolation to estimate its median
Answer
To estimate the median using linear interpolation, we first identify the cumulative frequency. The total number of commuters is 120, so the median is at the 60th position. By calculating the cumulative frequencies:
- 0-9: 10
- 10-19: 29 (10 + 19)
- 20-29: 72 (29 + 43)
The median falls in the 20-29 category. To interpolate:
y = L + \left(\frac{ \frac{N}{2}-F}{f} \right) \times c
Where:
- L = lower boundary of median class = 19.5
- N = total frequency = 120
- F = cumulative frequency of the class before the median = 29
- f = frequency of median class = 43
- c = class width = 10
Evaluating, we find:
[ y = 19.5 + \left(\frac{60-29}{43}\right) \times 10 \approx 26.7 ]
Step 3
Estimate the mean and the standard deviation of this distribution.
Answer
To calculate the mean () and standard deviation (\sigma), we can utilize the provided sums:
-
Mean: [ \mu = \frac{\sum fx}{N} = \frac{3550}{120} \approx 29.58 ]
-
Variance: [ \sigma^2 = \frac{\sum f x^2}{N} - \mu^2 = \frac{138020}{120} - (29.58)^2 \approx 16.56 ]
-
Standard Deviation: [ \sigma \approx \sqrt{\sigma^2} \approx 4.07 ]
Step 4
Evaluate this coefficient for this distribution.
Answer
Using the formula for skewness:
[ \text{Skewness} = \frac{3(\text{mean} - \text{median})}{\text{standard deviation}} ] Substituting the values:
- mean = 29.58
- median = 26.7
- standard deviation = 4.07
[ \text{Skewness} \approx \frac{3(29.58 - 26.7)}{4.07} \approx 0.520 ]
Step 5
State whether or not the value of your coefficient is consistent with your description in part (a). Justify your answer.
Answer
Yes, the value of the skewness coefficient is consistent with the description in part (a). The positive value of the skewness (0.520) indicates the distribution is positively skewed, aligning with the observation of more shorter commutes.
Step 6
State, with a reason, whether you should use the mean or the median to represent the data in this distribution.
Answer
The median should be used to represent the data in this distribution because the data is positively skewed. The median is less affected by outliers and extreme values, providing a more accurate measure of central tendency.
Step 7
State the circumstance under which it would not matter whether you used the mean or the median to represent a set of data.
Answer
It would not matter whether to use the mean or median if the data is symmetrical or uniformly distributed, as both measures of central tendency would provide a similar representation of the data.