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An estate agent is studying the cost of office space in London - Edexcel - A-Level Maths Statistics - Question 2 - 2017 - Paper 1
Question 2
An estate agent is studying the cost of office space in London. He takes a random sample of 90 offices and calculates the cost, £x per square foot. His results are g... show full transcript
Worked Solution & Example Answer:An estate agent is studying the cost of office space in London - Edexcel - A-Level Maths Statistics - Question 2 - 2017 - Paper 1
Step 1
Calculate the width and height of the bar representing 20 ≤ x < 40
Answer
To find the width of the bar for the interval 20 ≤ x < 40, we note that the range is from 20 to 40, giving:
Width = 40 - 20 = 20 units.
Since the area of the bar representing the frequency of offices is equal to 16 cm² for 32 offices, we set up the equation:
Area = width × height = 16 cm²
Thus, 20 cm × height = 16 cm²,
Solving for height, we find:
Height = ( \frac{16}{20} = 0.8 : \text{cm} ).
Step 2
Use linear interpolation to estimate the median cost.
Answer
To estimate the median cost, we find the cumulative frequency. The total number of offices is 90, so the median position is ( \frac{90}{2} = 45 ). The cumulative frequency leads us to:
- For 20 ≤ x < 40: Cumulative Frequency = 12
- For 40 ≤ x < 45: Cumulative Frequency = 12 + 13 = 25
- For 45 ≤ x < 50: Cumulative Frequency = 25 + 25 = 50
The median falls within the interval of 45 ≤ x < 50. Using linear interpolation between the midpoints of the cumulative frequencies:
Median = 45 + ( \frac{45 - 25}{50 - 25} \times (47.5 - 45) = 45 + 0.8 = 45.8 ).
Step 3
Estimate the mean cost of office space for these data.
Answer
To calculate the mean cost, use the formula:
( \text{Mean} = \frac{\sum (f \cdot y)}{\sum f} )
Calculating ( \sum (f \cdot y) ):
- For 20 ≤ x < 40: ( 12 \cdot 30 = 360 )
- For 40 ≤ x < 45: ( 13 \cdot 42.5 = 552.5 )
- For 45 ≤ x < 50: ( 25 \cdot 47.5 = 1187.5 )
- For 50 ≤ x < 60: ( 32 \cdot 55 = 1760 )
- For 60 ≤ x < 80: ( 8 \cdot 70 = 560 )
Thus, ( \sum (f \cdot y) = 360 + 552.5 + 1187.5 + 1760 + 560 = 3920 )
Now, ( \text{Mean} = \frac{3920}{90} \approx 43.56 ).
Step 4
Estimate the standard deviation for these data.
Answer
The standard deviation (SD) is calculated using the following formula:
( SD = \sqrt{\frac{\sum (f \cdot (y - \text{mean})^2)}{\sum f}} )
First, we find the variance:
Variance = ( \frac{(12(30-43.56)^2 + 13(42.5-43.56)^2 + 25(47.5-43.56)^2 + 32(55-43.56)^2 + 8(70-43.56)^2)}{90} )
Calculate the terms:
- For 20 ≤ x < 40: ( 12(30-43.56)^2 = 12(182.5936) )
- For 40 ≤ x < 45: ( 13(42.5-43.56)^2 = 13(1.0244) ) ... and so on for other intervals.
Finally, obtain ( SD = awrt 10.3 ).
Step 5
Describe, giving a reason, the skewness.
Answer
The data shows a slightly negative skewness since the mean (approximately 43.56) is less than the median (approximately 45.8). This indicates that there are a few lower-cost offices pulling the mean down, leading to a leftward tail in the distribution.
Step 6
With reference to your answer to part (e), comment on Rika’s suggestion.
Answer
Rika's suggestion of a normal distribution may not be entirely accurate considering the left skew observed. The data appears to be skewed rather than symmetrical, implying that a normal distribution might not fit well unless adjusted for the skewness.
Step 7
Use Rika’s model to estimate the 80th percentile of the cost of office space in London.
Answer
To find the 80th percentile using Rika's normal distribution (mean = £50, SD = £10):
Using the Z-score corresponding to 80th percentile, Z = 0.8416:
80th percentile = ( \text{Mean} + Z \cdot \text{SD} = 50 + 0.8416 \cdot 10 \approx 58.42 ).