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  3. Statistical Thinking in Python (Part 2)

Exercise

Bootstrap replicates of the mean and the SEM

In this exercise, you will compute a bootstrap estimate of the probability density function of the mean annual rainfall at the Sheffield Weather Station. Remember, we are estimating the mean annual rainfall we would get if the Sheffield Weather Station could repeat all of the measurements from 1883 to 2015 over and over again. This is a probabilistic estimate of the mean. You will plot the PDF as a histogram, and you will see that it is Normal.

In fact, it can be shown theoretically that under not-too-restrictive conditions, the value of the mean will always be Normally distributed. (This does not hold in general, just for the mean and a few other statistics.) The standard deviation of this distribution, called the standard error of the mean, or SEM, is given by the standard deviation of the data divided by the square root of the number of data points. I.e., for a data set, sem = np.std(data) / np.sqrt(len(data)). Using hacker statistics, you get this same result without the need to derive it, but you will verify this result from your bootstrap replicates.

The dataset has been pre-loaded for you into an array called rainfall.

Instructions

100 XP
  • Draw 10000 bootstrap replicates of the mean annual rainfall using your draw_bs_reps() function and the rainfall array. Hint: Pass in np.mean for func to compute the mean.
    • As a reminder, draw_bs_reps() accepts 3 arguments: data, func, and size.
  • Compute and print the standard error of the mean of rainfall.
    • The formula to compute this is np.std(data) / np.sqrt(len(data)).
  • Compute and print the standard deviation of your bootstrap replicates bs_replicates.
  • Make a histogram of the replicates using the normed=True keyword argument and 50 bins.
  • Hit submit to see the plot!