1. Interpreting CIs and technical conditions
By using the bootstrapped p-hat values in the previous exercises, you were able to find two different intervals for the true parameter of interest, the proportion of the population who will vote for candidate X.
2. Creating CIs
The first interval was based on the empirical rule, which extended two standard errors from the sample proportion to create the interval.
The second interval used the natural variability of the resampled p-hat values to characterize the distance from the true parameter. Remember, as in previous exercises, the quantile function finds the value which gives 2 (point) 5% of the bootstrapped proportions below and the value which gives 97 (point) 5% of the bootstrapped proportions below.
We can see that the two intervals are different because they used different methods, but they are consistent because they are based on the same observations.
3. Motivating CIs
Remember that the goal is to find an interval estimate of the parameter: the true proportion who will vote for candidate X, when the only information is the sample, which gives the proportion of individuals in the data who will vote for candidate X.
Ideas about the variability of p-hat-star are used to estimate how far the observed p-hat was from the population proportion.
Because we don't know whether the sample is close to the population or far from it, we don't know whether the confidence interval actually captures the true parameter. To that end, we interpret the interval using a confidence percentage.
4. Interpreting the CIs
That is, we say we are 95% confident that the true proportion of people planning to vote for candidate X is between 0 (point) 536 and 0 (point) 864.
5. Technical conditions
In this chapter, we have provided two ways for creating bootstrap confidence intervals.
These methods work for any statistic and parameter, as long as the following technical conditions hold: 1) the distribution of the statistic is reasonably symmetric and bell-shaped and 2) the sample size is reasonably large
A plot of the bootstrap p-hat-star values will give a good indication for whether the technical conditions are valid.
Both the bootstrap t-confidence interval and the bootstrap percentile interval measured the variability of the resampled proportions. Both methods will be used in later courses, along with more advanced methods, to find intervals for other parameters.
6. Let's practice!
OK, now it's your turn to practice what you've learned.