Interpret the Slope and the Y-Intercept of the Least-Squares Regression $\hat { y } = \beta _ { 0 } + \beta _ { 1 } x$

question 133

Multiple Choice

Interpret the Slope and the y-intercept of the Least-Squares Regression Line
-A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple linear regression model, $\hat { y } = \beta _ { 0 } + \beta _ { 1 } x$ , where $y =$ appraised value of the house (in \$thousands) and $x =$ number of rooms. Using data collected for a sample of $n = 74$ houses in East Meadow, the following results were obtained:
$$\begin{array} { l } \hat { y } = 74.80 + 19.72 \mathrm { x } \\\mathrm { s } \beta = 71.24 , \mathrm { t } = 1.05 \text { (for testing } \beta 0 \text { ) } \\\mathrm { s } \beta = 2.63 , \mathrm { t } = 7.49 \text { (for testing } \beta _ { 1 } \text { ) } \\\mathrm { SSE } = 60,775 , \mathrm { MSE } = 841 , \mathrm {~s} = 29 , \mathrm { r } ^ { 2 } = 0.44\end{array}$$
Range of the x-values: $5 - 11$
Range of the $y$ -values: $160 - 300$
Give a practical interpretation of the estimate of the $y$ -intercept of the least squares line.

• A) There is no practical interpretation, since a house with 0 rooms is nonsensical.
• B) For each additional room in the house, we estimate the appraised value to increase \(\$ 74,800\).
• C) For each additional room in the house, we estimate the appraised value to increase \(\$ 19,720\).
• D) We estimate the base appraised value for any house to be \(\$ 74,800\).

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Definitions:

Exponential Decay

A process in which the value of a quantity decreases at a rate proportional to its current value.

Compound Interest

The addition of interest to the principal sum of a loan or deposit, where the interest that has been added also earns interest.

Exponential Growth

Exponential growth describes a process that increases in quantity at a consistent rate relative to its current value, often seen in populations or investments.

Arithmetic Operations

Basic mathematical operations which include addition, subtraction, multiplication, and division.