Simple Linear Regression: Coefficients, Interpretation, and Prediction

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What is the equation for a simple linear regression model?

It is written $\hat{y} = a + bx$, where $\hat{y}$ is the predicted response, $a$ is the intercept, and $b$ is the slope.

How is the least-squares slope calculated from paired data?

$b = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}$.

How is the intercept calculated from the slope and data means?

$a = \bar{y} - b\bar{x}$.

What argument order do Excel's SLOPE and INTERCEPT functions use?

Both take the $y$-data range first and the $x$-data range second.

When is a regression intercept meaningful in context?

When $x=0$ is meaningful for the situation and supported by the observed data range.

Quiz questions

1. In a simple linear regression model, what does the coefficient $b$ represent?Answer: The slope of the fitted line.
2. Which expression gives the least-squares slope for $n$ paired observations?Answer: $b = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}$
3. If monthly revenue is modeled as depending on monthly advertising expenditure, which variable is the predictor $x$?Answer: Monthly advertising expenditure.
4. For an advertising-and-revenue model with intercept $9{,}376.7$ and slope $61.8$, which equation is correct?Answer: $\hat{y} = 9{,}376.7 + 61.8x$
5. An advertising-and-revenue model is $\hat{y} = 9{,}376.7 + 61.8x$, with both variables measured in thousands of dollars. What revenue does it predict when advertising is $100$ thousand dollars?Answer: $15{,}556.7$ thousand dollars.