In your initial post, address the following items: Module Two Discussion: The Central Limit Theorem

This notebook contains the step$-$by$-$step directions for your Module Two discussion. It is very important to run through the steps in order. Some steps depend

on the outputs of earlier steps. Once you have completed the steps in this notebook, be sure to answer the questions about this activity in the Discussion for

on the outpul this modul.

Reminder: If you have not already reviewed the discussion prompt, please do so before beginning this activity. That will give you an idea of the questions you

will need to answer with the outputs of this script.

Initial post $($due Thursday$)$

Step $1$: Generating population data

This block of Python code will generate unique TPCP population data of size $500$ observations. You will use this data set in this week's discussion. The numpy

module in Python can be used to create datasets with a skewed distribution by randomly generating data from a gamma distribution. You do not need to know

what a gamma distribution is or how a dataset is drawn from it$.$ The dataset will be saved in a Python dataframe that you will use in later calculations.

Click the block of code below and hit the Run button above.

In $[1]$:

import pandas as pd import matplotlib.pyplot as plt

import numpy as np import scipy.stats as st

# use gamma distribution to randomly generate $500$ observations

shape, scale $=1\mathrm{.}95,2\mathrm{.}5$ tpcp $=100*$ np$.$random.gamma$($shape$,$ scale, $500)$

# pandas library can be used to convert the array into a dataframe of rounded figures with the column name TPCP$.$

tpcp$\_$df $=$ pd$.$ DataFrame$($tpcp$,$ columns $\text{'}$TPCP$\text{'}])$

round$(0)$

# print the dataframe to see the first $5$ and last $5$ observations $($note that the index of dataframe starts at $0).$

print$("$TPCP data frame $"$

In the Python script, you created a histogram for the dataset

generated in Step $1.$ Check to make sure that this data

distribution is skewed and included in your attachment. See

Step $2$ in the Python script.

What is the mean of the TPCP population data? See Step $3$ in

the Python script.

In the Python script, you selected a random sample with

replacement, of size $50($note that this is a sufficiently large

sample$),$ from the TPCP population. What is the mean of your

random sample? Does this sample mean closely approximate

the TPCP population mean? See Step $4$ in the Python script.

You also selected $1,000$ random samples of size $50$ and

calculated the mean of each sample. Then you stored those

means into a dataframe. Check to make sure the output of this

step is in your attachment. See Step $5$ in the Python script.

Review the plotted data distribution for these $1,000$ means.

Does this approximate a Normal distribution? Does this

confirm the first part of the central limit theorem? Why or why

not? See Step $6$ in the Python script.

What is the "grand" mean and standard deviation of these

$1,000$ means? Does the grand mean closely approximate $($on a

relative basis$)$ the mean of the original distribution? Does this

confirm the second part of the central limit theorem? Why or

why not? See Step $7$ in the Python script.