University of California, Berkeley
DATA MISC
1 Homework 9: Central Limit Theorem
Reading: * Why the mean matters
Please complete this notebook by filling in the cells provided. Before you begin, execute the
foll
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University of California, Berkeley
DATA MISC
1 Homework 9: Central Limit Theorem
Reading: * Why the mean matters
Please complete this notebook by filling in the cells provided. Before you begin, execute the
following cell to load the provided tests. Each time you start your server, you will need to execute
this cell again to load the tests.
Homework 9 is due Thursday, 11/1 at 11:59pm. You will receive an early submission bonus
point if you turn in your final submission by Wednesday, 10/31 at 11:59pm. Start early so that
you can come to office hours if you’re stuck. Check the website for the office hours schedule. Late
work will not be accepted as per the policies of this course.
Directly sharing answers is not okay, but discussing problems with the course staff or with
other students is encouraged. Refer to the policies page to learn more about how to learn cooperatively.
For all problems that you must write our explanations and sentences for, you must provide your answer in the designated space. Moreover, throughout this homework and all future
ones, please be sure to not re-assign variables throughout the notebook! For example, if you use
max_temperature in your answer to one question, do not reassign it later on.
In [1]: # Don't change this cell; just run it.
import numpy as np
from datascience import *
# These lines do some fancy plotting magic.
import matplotlib
%matplotlib inline
import matplotlib.pyplot as plt
plt.style.use('fivethirtyeight')
import warnings
warnings.simplefilter('ignore', FutureWarning)
from client.api.notebook import Notebook
ok = Notebook('hw09.ok')
_ = ok.auth(inline=True)
=====================================================================
Assignment: Homework 9: Central Limit Theorem
1
OK, version v1.12.5
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1.1 1. The Bootstrap and The Normal Curve
In this exercise, we will explore a dataset that includes the safety inspection scores for restaurants
in the city of Austin, Texas. We will be interested in determining the average restaurant score for
the city from a random sample of the scores; the average restaurant score is out of 100. We’ll compare two methods for computing a confidence interval for that quantity: the bootstrap resampling
method, and an approximation based on the Central Limit Theorem.
In [2]: # Just run this cell.
pop_restaurants = Table.read_table('restaurant_inspection_scores.csv').drop(5,6)
pop_restaurants
Often it is impossible to find complete datasets like this. Imagine we instead had access only to
a random sample of 100 restaurant inspections, called restaurant_sample. That table is created
below. We are interested in using this sample to estimate the population mean.
Question 3 Complete the function bootstrap_scores below. It should take no arguments.
It should simulate drawing 5000 resamples from restaurant_sample and computing the mean
restaurant score in each resample. It should return an array of those 5000 resample means.
In [8]: def bootstrap_scores():
resampled_means = make_array()
for i in range(5000):
resampled_mean = np.mean(restaurant_sample.sample().column(3))
resampled_means = np.append(resampled_means, resampled_mean)
return resampled_means
resampled_means = bootstrap_scores()
resampled_means
Question 4 Compute a 95 percent confidence interval for the average restaurant score using the
array resampled_means.
In [11]: lower_bound = percentile(2.5, resampled_means)
upper_bound = percentile(97.5, resampled_means)
print("95% confidence interval for the average restaurant score, computed by bootstrap
95% confidence interval for the average restaurant score, computed by bootstrapping:
( 90.98 , 93.56 )
Question 5 Does the distribution of the resampled mean scores look normally distributed? State
"yes" or "no" and describe in one sentence why you would expect that result.
Yes, since the central limit theorem states that the distribution of sample averages tend to be
normally distributed
6
Question 6 Does the distribution of the sampled scores look normally distributed? State "yes"
or "no" and describe in one sentence why you should expect this result.
Hint: Remember that we are no longer talking about the resampled means!
No, since the sampled scores are distributed like the population scores , and the population
scores are not normally distributed.
For the last question, you’ll need to recall two facts. 1. If a group of numbers has a normal distribution, around 95% of them lie within 2 standard deviations of their mean. 2. The Central Limit
Theorem tells us the quantitative relationship between the following: * the standard deviation of
an array of numbers. * the standard deviation of an array of means of samples taken from those
numbers.
Question 7 Without referencing the array resampled_means or performing any new simulations,
calculate an interval around the sample_mean that covers approximately 95% of the numbers in
the resampled_means array. You may use the following values to compute your result, but you
should not perform additional resampling - think about how you can use the CLT to accomplish
this.
In [12]: sample_mean = np.mean(restaurant_sample.column(3))
sample_sd = np.std(restaurant_sample.column(3))
sample_size = restaurant_sample.num_rows
mean_sd = sample_sd / sample_size**0.5
lower_bound_normal = sample_mean - 2 * mean_sd
upper_bound_normal = sample_mean + 2 * mean_sd
print("95% confidence interval for the average restaurant score, computed by a normal
95% confidence interval for the average restaurant score, computed by a normal approximation:
( 90.9258714979737 , 93.6341285020263 )
This confidence interval should look very similar to the one you computed in Question 4.
1.2 2. Testing the Central Limit Theorem
The Central Limit Theorem tells us that the probability distribution of the sum or average of a
large random sample drawn with replacement will be roughly normal, regardless of the distribution
of the population from which the sample is drawn.
That’s a pretty big claim, but the theorem doesn’t stop there. It further states that the standard
deviation of this normal distribution is given by
sd of the original distribution
psample size
In other words, suppose we start with any distribution that has standard deviation x, take a
sample of size n (where n is a large number) from that distribution with replacement, and compute
the mean of that sample. If we repeat this procedure many times, then those sample means will
have a normal distribution with standard deviation pxn.
That’s an even bigger claim than the first one! The proof of the theorem is beyond the scope of
this class, but in this exercise, we will be exploring some data to see the CLT in action.
