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

question 25

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:
$$\begin{array} { l } \hat { y } = 74.80 + 22.75 x \\\mathrm {~s} \beta = 71.24 , t = 1.05 \text { (for testing } \beta _ { 0 } \text { ) } \\\mathrm { s } \beta = 2.63 , \mathrm { t } = 7.49 \text { (for testing } \beta _ { 1 } \text { ) } \\\mathrm { SSE } = 60,775 , \mathrm { MSE } = 841 , \mathrm {~s} = 29 , \mathrm { r } ^ { 2 } = .44\end{array}$$
Range of the x-values: $5 - 11$
Range of the $y$ -values: $160 - 300$
Give a practical interpretation of the estimate of the slope of the least squares line.

• A) For each additional room in the house, we estimate the appraised value to increase \(\$ 22,750\).
• B) For each additional room in the house, we estimate the appraised value to increase \(\$ 74,800\).
• C) For each additional dollar of appraised value, we estimate the number of rooms in the house to increase by \(22.75\) rooms.
• D) For a house with 0 rooms, we estimate the appraised value to be \(\$ 74,800\).

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Definitions:

Boxplot

A graphical representation that displays the distribution of a dataset based on a five-number summary: minimum, first quartile, median, third quartile, and maximum.

Histogram

A graphical representation of the distribution of numerical data through bars of various heights.

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Measurements reflecting the degree of heat present in a substance or environment, often expressed in degrees Celsius, Fahrenheit, or Kelvin.

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The amount of money or compensation paid to an employee for work performed during a week, often expressed as an annual figure divided by 52 weeks.