Question

A lazy employee at an off-campus second-hand bookstore has decided that, rather than go searching through the book to find the correct price, he will start pricing the books in relation to how thick they actually are. A sample of 10 books, ranging in thickness from 1 cm to 6 cm, gave the following results for thickness (x, in cm) and price (y, in $): Sxy = 87, Sxx = 24, Syy = 358.9, Σx = 30, and Σy = 261. What is the predicted price of a textbook that is 1 cm thick?

Answer 1

The predicted price for of a textbook that is 1 cm thick is
`\= y =` $18.86

Explanation:

From the question we are told that

The sample size is n = 10

The maximum thickness is
`t_(max) = 6 cm`

The minimum thickness is
`t_(min) = 1 cm`

The sum Σx = 30

Sxy = 87

Syy = 358.9

Σy = 261

The mean thickness is

`\= x = (\sum x)/(n)`

Substituting value

`\= x = (30)/(10)`

`\= x = 3`

The mean price of the book is

`\= y = (\sum y)/(n)`

Substituting value

`\= y = (261)/(10)`

`\= y =26.1`

Generally the least square regression equation is mathematically represented as

`\r y = b_o + b_1 x`

`\r y` is the predicted price

Where
`b_1` is a constant evaluated as

`b_o = (SS_(xy))/(SS_(xx))`

Substituting value

`b_o = (87)/(24)`

`b_o = 3.625`

At mean price and thickness The least square regression equation becomes

`\= y = b_o + b_1 \= x`

i.e
`\r y = \= y , x= \= x`

Substituting value

`26.1 = b_o + 3.625 * 3`

=>
`b_o = 15.23`

For a thickness of 1 cm the predicted price is

`\= y = 15.23 + (3.625) *1`

`\= y =` $18.86