Inferences about regression

The total length of the videos in this section is approximately 21 minutes. You will also spend time answering short questions while completing this section.

You can also view all the videos in this section at the YouTube playlist linked here.

Comparison to t-tests

RegressionInferences.1.ComparingtoT-Tests.mp4

Question 1: If we run a simulation that repeatedly samples data from a linear regression model with a certain intercept and slope, which set of values must be different for each data set, X or Y?

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Y. The X values can be arbitrarily chosen for a simulation, and then can remain the same for each data set created in the simulation. The linear regression model describes the distribution of Y values for particular values of X. So, X divides the units into groups, much like the grouping variable in a t-test.

Distribution of estimate of slope

RegressionInferences.2.DistributionsBetaHat.mp4

Question 2: Which of the following result in a better estimate of the slope?

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X values that are farther apart from one another

Question 3: Which of the following influences the variance of the estimate of the slope? Check all that apply.

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All three. When the residual variance is big, the points are further from the line, and so we are less sure of our estimate of the slope (and the variance of β₁-hat increases). When the X values are spread far apart, we have more information about the slope of the line than we would if all the points had similar values of X (and the denominator of the variance of β₁-hat increases, so that the variance of β₁-hat decreases). As the sample size increases, we have more information about the slope (and the summation in the denominator of the variance of β₁-hat increases, so that the variance of β₁-hat decreases). If the sample size was equal to infinity, the variance of β₁-hat would be 0, because we would have information about all the units in the population, and our estimate of β₁-hat would be exactly equal to β₁.

Hypothesis tests with linear regression

Note: there is a typo in the following video. At 5:24, I write the variance of the intercept estimate incorrectly. I omitted a "1/n" term. The variance of the intercept should be:

 sigma^2 * (1/n + xbar^2 / (sum of squared difference in X))

RegressionInferences.3.HypTests.mp4

Question 4: Which is usually more useful to estimate, the slope or the intercept?

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The slope. We usually don't particularly care what the mean of Y is when X is 0 (this is the intercept). We are much more interested in how the mean of Y changes when X increases by specified amount (this is the slope).

You've reached the end.