Question

A linear regression model describing the relationship between the carat weight and price of very high quality diamonds is summarized below.

r= 0.9693
y = 20312x - 5939

A diamond seller lists a very high quality diamond weighing 0.8 carats at a price of $9,999. Does this model over- or under-predict the price of this diamond?

a. The model over-predicts the price of this diamond because the residual is positive.
b. The model under-predicts the price of this diamond because the residual is positive.
c. The model over-predicts the price of this diamond because the residual is negative.
d. The model under predicts the price of this diamond because the residual is negative.
e. We do not have enough information to answer this question.

Answer 1

The correct option is c.

Explanation:

The regression line representing the price of very high quality diamonds is:

`y = 20312\cdot x - 5939`

It is provided that a diamond seller lists a very high quality diamond weighing 0.8 carats at a price of $9,999.

Compute the predicted price of the diamond weighing 0.8 carats as follows:

`\hat y = 20312\cdot x - 5939`

`=( 20312* 0.8) - 5939\\\\=10310.60`

The predicted price of the diamond weighing 0.8 carats is $10,310.60.

In regression, the difference amid the observed-value of the dependent-variable (y) and the predicted-value (
`\hat y`) is known as the residual (e).

Compute the residual value as follows:

`e=y-\hat y`

`=9999-10310.60\\\\=-311.60`

The residual value is -$311.60.

Thus, it can be concluded that the model over-predicts the price of this diamond because the residual is negative.

The correct option is c.