- Perform a test for significance of slope and interpret the results
- Check the conditions that are necessary to perform a test for significance of slope
What’s in the Price of a House?
House prices fluctuate over time and are impacted by several variables. There are both internal variables (characteristics of the house) and external variables (the economy) that can impact house prices. It is even possible for two identical houses to sell at different prices.
Previously, you learned that there are several conditions that must be met in order to analyze data using linear regression. Similarly, in order to conduct a hypothesis test for the significance of slope, you will need to verify those same conditions.
Recall that when a linear regression is appropriate, the value of the residuals will be randomly scattered around [latex]0[/latex]. In other words, some residuals will be positive (observed value above the line on the residual plot) and some will be negative (observed value below the line on the residual plot). We should not see a systematic pattern (e.g., all above in order, then all below in order).
We should worry about the appropriateness of the model if we notice the following red flags:
- Red Flag 1: The trend in the scatterplot is nonlinear, indicating that the relationship between the explanatory variable and the response variable is not modeled well by a line. The residuals tend to have a pattern.
- Red Flag 2: The observed values are further and further away from the line of best fit for a portion of the data. In other words, the errors are not consistent for all values of the explanatory variable. The size of the residuals tends to increase or decrease as the value of the explanatory variable increases. When this happens, it can be hard to get a handle on the accuracy of the model because the standard deviation of the residuals is not constant over the values of the independent variable.
Formally, to conduct a hypothesis test for the slope, we require:
- A simple random sample
- For any given value of [latex]x[/latex], the residuals have a mean of [latex]0[/latex] (Red Flag 1)
- The standard deviation of the residuals is the same at all values of (Red Flag 2)
- For any given value of [latex]x[/latex], the residuals follow a normal distribution.