- A large company rents shops in different parts of the country. A random sample of 10 shops was taken and the floor area, \(x\) in \(10 \mathrm {~m} ^ { 2 }\), and the annual rent, \(y\) in thousands of dollars, were recorded.
The data are summarised by the following statistics
$$\sum x = 900 \quad \sum x ^ { 2 } = 84818 \quad \sum y = 183 \quad \sum y ^ { 2 } = 3434$$
and the regression line of \(y\) on \(x\) has equation \(y = 6.066 + 0.136 x\)
- Use the regression line to estimate the annual rent in dollars for a shop with a floor area of \(800 \mathrm {~m} ^ { 2 }\)
- Find \(\mathrm { S } _ { y y }\) and \(\mathrm { S } _ { x x }\)
- Find the product moment correlation coefficient between \(y\) and \(x\).
An 11th shop is added to the sample. The floor area is \(900 \mathrm {~m} ^ { 2 }\) and the annual rent is 15000 dollars.
- Use the formula \(\mathrm { S } _ { x y } = \sum ( x - \bar { x } ) ( y - \bar { y } )\) to show that the value of \(\mathrm { S } _ { x y }\) for the 11 shops will be the same as it was for the original 10 shops.
- Find the new equation of the regression line of \(y\) on \(x\) for the 11 shops.
The company is considering renting a larger shop with area of \(3000 \mathrm {~m} ^ { 2 }\)
- Comment on the suitability of using the new regression line to estimate the annual rent. Give a reason for your answer.