Variability in p-hat

1. Variability in p-hat

Remember the goal of creating a confidence interval is to find a range of plausible values for the true population parameter.

2. How far are the data from the parameter?

The observed statistic is a good estimate, but it is impossible to know whether it is one of the p-hat values,

3. How far are the data from the parameter?

which is very close to the true parameter

4. How far are the data from the parameter?

or whether it is one of the p-hat values, which is in the tail of the distribution of observed statistics. We won't know how far away the statistic is from the parameter just by looking at the sample of data. The variability of the p-hat statistics gives a measure for how far apart any given observed p-hat and the parameter are expected to be.

5. Standard error of p-hat

Note that bootstrapping, or resampling with replacement the same number of observations as were in the original sample, provided approximately the same standard error as sampling many p-hat values from the population. The standard error is measured by the width of the distribution and here we can see that the green bootstrap distribution is approximately the same width as the black distribution, which measured the true variability of p-hat values from the population. Lucky for us that bootstrapping works because in real life we only have one dataset! We can use the value of the standard error to count the number of sample p-hats, which are close to the parameter.

6. Empirical rule

It turns out that because the distribution of p-hat values is symmetric and bell shaped,

7. Empirical rule

roughly 95% of samples will produce p-hats that are within two standard errors of the center. This idea is called the empirical rule and comes from theory which describes bell-shaped distributions.

8. Empirical rule

Note that in the exercises, we refer to the intervals created from the empirical rule as t-intervals. The "t" is simply the label that statisticians use and we provide that label here for you in case you want to learn more about these types of intervals in future courses. Here, in red, we see all the sample p-hat values that are close to the true parameter. If our data is one of the red dots, our confidence interval will correctly capture the true parameter.

9. Let's practice!

OK, now it's your turn to practice what you've learned.

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