Course Outline
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segmentGetting Started (Don't Skip This Part)
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segmentStatistics and Data Science: A Modeling Approach
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segmentPART I: EXPLORING VARIATION
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segmentChapter 1 - Welcome to Statistics: A Modeling Approach
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segmentChapter 2 - Understanding Data
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segmentChapter 3 - Examining Distributions
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segmentChapter 4 - Explaining Variation
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segmentPART II: MODELING VARIATION
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segmentChapter 5 - A Simple Model
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segmentChapter 6 - Quantifying Error
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segmentChapter 7 - Adding an Explanatory Variable to the Model
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segmentChapter 8 - Digging Deeper into Group Models
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segmentChapter 9 - Models with a Quantitative Explanatory Variable
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segmentPART III: EVALUATING MODELS
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segmentChapter 10 - The Logic of Inference
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segmentChapter 11 - Model Comparison with F
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segmentChapter 12 - Parameter Estimation and Confidence Intervals
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12.4 Interpreting the Confidence Interval
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segmentChapter 13 - What You Have Learned
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segmentFinishing Up (Don't Skip This Part!)
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segmentResources
list High School / Advanced Statistics and Data Science I (ABC)
12.4 Interpreting the Confidence Interval
Now that we have spent some time constructing confidence intervals, it is important to pause and think about what a confidence interval means, and how it fits with other concepts we have studied so far.
Confidence Intervals Are About the DGP
One common misconception about confidence intervals is that they define lower and upper cutoffs for where .95 of the
But that was just a method for calculating the interval, not a definition of what the interval actually refers to. It is important to remember that the concept of confidence interval was developed by mentally moving the sampling distribution of
If we did want to know the range of possible sample
Error in an Estimate
The
But being the best doesn’t mean it’s right. This estimate is almost certainly wrong. It might be too low or it might be too high, but we don’t know which way it is wrong. And to make matters worse, it’s hard to tell whether the estimate is correct or not, or how far off it is from the true DGP, because we don’t really know what the true
The confidence interval provides us with a way of addressing this problem. It tells us how wrong we could be, or put another way, how much error there might be in our estimate.
If the confidence interval is relatively wide given the situation, as it is in the tipping study, we would be saying something like, “the estimated effect of adding a smiley face to the check is $6.05. But there is a lot of error in the estimate. The true effect could be as low as 0 or slightly below that, or as high as $13.”
The width of the confidence interval (CI) tells us what the true
It is important to note that when we talk about the error in an estimate we are using the term error to mean something a little different than we have learned up to now. Previously, when we developed the concept of error (as in DATA = MODEL + ERROR), we were referring to the gap between the predicted tip for each table based on a model, and the actual tip left by that table. The errors were the individual residuals for each table.
When we think about error around a parameter estimate though, we’re not thinking about individual tables any more. A single table can’t have a
Because we generally don’t know what the true
In the case of the tipping experiment, we started with a point estimate of the
What Does the 95% Mean?
One question you might have is this: what does it mean to have 95% confidence?
Let’s start by explaining what it does not mean. It does not mean that there is a .95 probability that the true
One reason they will correct you is that
The other reason they will correct you is that there isn’t actually a .95 chance that the
Because of this issue, someone (actually, a mathematician named Jerzy Neyman, in 1937) came up with the idea of saying “95% confident” instead of “95% probable.” Our guess is that all the statisticians and mathematicians breathed a sigh of relief over this.
When you construct a 95% confidence interval, therefore, you are saying that you are 95% confident (alpha = .05) that the true
Confidence Intervals and Model Comparison
We have now used the sampling distribution of
The confidence interval provides us with a range of models of the DGP (i.e., a range of possible
We would reject any values of
Using A Confidence Interval to Evaluate the Empty Model |
Using a Hypothesis Test to Evaluate the Empty Model |
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In the right panel of the figure above, the model comparison (or hypothesis testing) approach considers just one particular model of the DGP, not a range of models. In this model, in which shuffle()
to mimic such a DGP, and built a sampling distribution centered at 0. We can see in the picture above that if such a DGP were true, our sample
We then used the sampling distribution as a probability distribution to calculate the probability of getting a sample
These two approaches – null hypothesis testing and confidence intervals – both provide ways of evaluating the empty model, and both lead us to the same conclusion in the tipping study: the empty model, where
If the 95% confidence interval does not include 0, then we would reject the empty model because we are not confident that
As another example, let’s consider a second tipping study done by another team of researchers. They got very similar results but this time, their
Original Study ( |
Second Study ( |
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We don’t really believe that the DGP has changed, so we wouldn’t say the
Let’s take a look at how the confidence interval might be different across these two studies.
Original Study ( |
Second Study ( |
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In the left panel of the figure the confidence interval (marked by the two red boxes) is centered around an assumed
In the right panel of the figure, we see what happened in the second study where the observed
It is also worth noting that we get a lot more information from the confidence interval than we do from the p-value. For example, in the original tipping study (where