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Unit 9

9.2 Confidence Intervals for the Slope of a Regression Model

3 min readapril 28, 2021

josh

Josh Argo


9.2: Confidence Intervals for the Slope of a Regression Model

Confidence Intervals

As previously stated, there are two kinds of inference:  prediction and testing claims. The first type of inference in regards to linear regression we are going to tackle is the prediction model:  confidence intervals. A confidence interval is a range of values that is almost certain (whatever percentage interval we construct) to contain our true value for the population.
In linear regression, our biggest interest is the slope of our regression line. While we can somewhat easily calculate the slope of our sample, we know that this slope is going to change as we add more data points and could change greatly if we were to add several more data points. So instead of just relying on our sample slope, it is a much more robust process to construct a confidence interval to find ALL possible values of our slope.
Point Estimate
The first part of our confidence interval is our point estimate. This is the exact slope of our sample data that can be calculated using the methods discussed in Unit 2. This is the middle of our confidence interval and our starting point. From there, we are going to add and subtract our margin of error to give us a “buffer zone” around our sample prediction.
Margin of Error
Our margin of error for our confidence interval is calculated by using the appropriate t score and the standard deviation of the residuals and standard deviation of the x values. 
Our t score is based on the confidence level and degrees of freedom as mentioned in Unit 7.
Our standard error can be calculated using the formula on the formula sheet (below).
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-R9bKdaBdUQpr.png?alt=media&token=b7f44a94-8d4e-41e0-8672-f48fdafc7a5d

Image Taken from Collegeboard

This formula is very cumbersome and I would always recommend using your graphing calculator to calculate your interval by selecting LinRegTInt and using your L1 and L2 for your sample data.

Conditions

As with every inference procedure we have covered, there are conditions for inference that must be met if we carry out a test or construct an interval:
Linear
The first and easiest condition to check is that the true relationship between our x and y variable appear to be linear. This can be confirmed by observing the residual plot and seeing that there is no real pattern in the residuals.
Standard Deviation of y 
The next condition that must be met is that the standard deviation of y must not change as x changes. In other words, our residual plot does not scatter more or less as we move down the x axis. Again, there is absolutely no pattern on the residual plot
Independence
Independence can be checked two ways: 
  1. Data was taken from a random sample or randomized experiment
  2. 10% condition (same as other inference procedures):  “It is reasonable to believe there is at least 10n… in our population”
Normal
As we have all figured out by now, everything hinges on the normal distribution in statistics. To show normal distribution for quantitative data, we use the Central Limit  Theorem, which states that our sample size is at least 30 (or our y values are approximately normal as is).
Once you have checked your conditions, you are good to go in using your sample data to construct and interpret a confidence interval for quantitative means (which I strongly recommend using some form of technology like a graphing calculator to do!
🎥  Watch: AP Stats Unit 9 - Inference for Slopes

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