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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
The first step is to determine the width of the bar. Given that the bar for 50 ≤ x < 60 is 2 cm wide, and knowing that all bars are proportional to their frequency, we calculate the factor.
For the bar representing 20 ≤ x < 40, the class width is: [ \text{Width} = 40 - 20 = 20 ext{ (units)} ] Thus, its area can be represented as: [ \text{Area} = \text{Width} \times \text{Height} = 16 \text{ cm}^2 ext{ (representing 32 offices)} ] Let h be the height of the bar. The relationship can be expressed as: [ 2h = 16 \Rightarrow h = 8 \text{ cm} ] Hence, the width is 4 cm and the height is 8 cm.
Step 2
Use linear interpolation to estimate the median cost.
Answer
The median is the value that separates the higher half from the lower half of the data set. Since there are 90 offices, the median will be the 45th office when arranged in ascending order.
Calculating cumulative frequencies:
- 20 ≤ x < 40 : 12
- 40 ≤ x < 45 : 13 (Cumulative: 25)
- 45 ≤ x < 50 : 25 (Cumulative: 50)
The 45th office falls into the interval 45 ≤ x < 50. To find the exact median value, we will use linear interpolation. The lower boundary (L) is 45, and the frequency (f) for the interval is 25. The cumulative frequency before this interval is 25.
The formula for linear interpolation is: [ \text{Median} = L + \left(\frac{\frac{n}{2} - CF}{f}\right) \times c \text{ (where } c \text{ is the class width)} ] Substituting in, we have: [ \text{Median} = 45 + \left(\frac{45 - 25}{25}\right) \times 5] [ = 45 + \left(\frac{20}{25}\right) \times 5] [ = 45 + 4 = 49] Thus, the estimated median cost is £49.
Step 3
Estimate the mean cost of office space for these data.
Answer
The mean can be calculated using the formula: [ \bar{x} = \frac{\sum f y}{n} ] Where (n) is the total number of offices (90), and (f) and (y) are the frequency and midpoint respectively.
Calculating (\sum f y): [ \sum f y = (12 \times 30) + (13 \times 42.5) + (25 \times 47.5) + (32 \times 55) + (8 \times 70) ] [ = 360 + 552.5 + 1187.5 + 1760 + 560 = 3920 ] Now, substituting into the mean formula: [ \bar{x} = \frac{3920}{90}= 43.56 ] Therefore, the estimated mean cost of office space is £43.56.
Step 4
Estimate the standard deviation for these data.
Answer
The standard deviation can be estimated using the formula: [ \sigma = \sqrt{\frac{\sum f y^2}{n} - (\bar{x})^2} ] Where we first need to calculate (\sum f y^2): [ \sum f y^2 = (12 \times 30^2) + (13 \times 42.5^2) + (25 \times 47.5^2) + (32 \times 55^2) + (8 \times 70^2) ] Calculating each term gives us: [ = 10800 + 11406.25 + 5625 + 9680 + 3920 = 39412.25 ] Now substituting into the standard deviation formula: [ \sigma = \sqrt{\frac{39412.25}{90} - (43.56)^2} ] Calculating: [ \sigma \approx \sqrt{438.026 - 1894.3536} \approx \sqrt{3750} \approx 61.25 ] Therefore, the estimated standard deviation is approximately £61.25.
Step 5
Describe, giving a reason, the skewness.
Answer
Skewness refers to the asymmetry of the distribution of values. By observing the calculated mean and median, where the mean (approximately £43.56) is less than the median (£49), it suggests a leftward skew.
This indicates that there are more lower values affecting the mean, thus resulting in a distribution that is not symmetrical.
Step 6
With reference to your answer to part (e), comment on Rika's suggestion.
Answer
Rika's suggestion assumes a normal distribution of office costs with a mean of £50 and standard deviation of £10. Given the left skew observed in the actual data, it indicates that Rika's model may not accurately represent the realities of the market. The actual data suggests there are many offices priced below the mean, leading to the conclusion that the cost of office space might not follow a normal distribution.
Step 7
Use Rika's model to estimate the 80th percentile of the cost of office space in London.
Answer
To determine the 80th percentile in a normal distribution, we can use the formula: [ P = \mu + z \cdot \sigma ] Where (\mu = 50) (mean), (\sigma = 10) (standard deviation), and (z) is the z-score corresponding to the 80th percentile (which is approximately 0.8416).
Substituting in the values: [ P_{80} = 50 + 0.8416 \cdot 10 \approx 58.416 ] Thus, Rika's model estimates that the 80th percentile cost of office space in London is approximately £58.42.