Question

Python 3.7 please help

please use central limit theory

In this problem you will verify the Central Limit Theorem (CLT) which states that averages, from repeated random samples of any distribution, follow a normal distribution 1. (5 points) Draw a random sample of 5,000 random numbers from a uniform distribution X ~U (20,80] and store them into a vector called xy and plot a histogram of these 5,000 numbers 2. (5 points) Draw a random sample of 5,000 random numbers from a uniform distribution X - U 20,80) and calculate the average of these numbers and assign it to the variable xbar. Print it on the screen. 3. (5. points) Draw 300 random samples of 5,000 random numbers each from a uniform distribution X-U (20,80) and for each one of the 300 random samples calculate the average of the 5,000 numbers in the respective sample and store those 300 averages in a vector called xbarv. 4. (5 points) The CLT says that the numbers stored in xbarv from the previous question follow a normal distribution. Can you see it by drawing a histogram of xbarv? 5. (5 points) Write a function that does all the steps from steps 3-4 so that the function returns the x barv vector. As input the function should require the number of random samples, the size of each random sample, the lower limit of the Uniform distribution and the upper limit of the uniform distribution, so that you can call the function as: xbarv =f_my CLT(nr Samples = 300, sample Size=5000, lower=20, upper=80) 6. (5 points) Test your function by calling it with: xbarv = my CLT(nr Samples=400, sample Size=10000, lower=20, upper=80) and making another histogram of this new xbarv. How does the histogram look now? Bell shaped

Answer 1

ANSWER:

Here, NumPy and matplotlib are used. so, we need it import first:

[23] 1 import numpy as np 2 import matplotlib.pyplot as plt

[] 1 # 21: 2 xv = np.random.uniform( 20, 80,5000)

[19] 1 # Q2: 2 xbar np.average(xv) 50.424024819160906 Cole Tort

1 # Q3: 2 # store average of each random dram in local var, and finally append that to xbarv 3 xbarv=[] 4 for i in range(300): 5 xv = np.random.uniform( 20,80,5000) 6 xbar = np.average (xv) 7 xbarv.append(xbar) Tada Ta

1 # 04: 2 # Bins size should be set comparable to data difference 3 # Using matploit lib to represent graph 4 5 plt.hist(xbarv, bins=np.arange(min(xbarv), max(xbarv)+0.09, 0.09)) 6 plt.title("histogram") 7 plt.show() histogram 40 35 30 25 20 15 10 5 49.2 49.4 49.6 49.8 50.0 50.2 50.4 50.6 50.8

4 1 # 25: 2 # merge the above code 3 def f_myCLT(nr Samples, sampleSize, lower, upper): xbarv=[] 5 for i in range(nr Samples): xv = np.random.uniform(lower, upper, nr Samples) 7 xbar = np.average (xv) xbarv.append(xbar) 9 return xbarv 8 Pare Toy

[104] 1 # 26: 2 # test code 3 xbarv = f_myCLT(400,10000, 20,80) 4 plt.hist(xbarv, bins=np.arange(min(xbarv), max(xbarv)+0.1, 0.1)) 5 plt.title("histogram") 6 plt.show() histogram 25 20 15 10 5 0 49 50 51 52

CODE:

import numpy as np

import matplotlib.pyplot as plt

# Q1:

xv = np.random.uniform(20,80,5000)

# Q2:

xbar = np.average(xv)

# Q3:

# store average of each random dram in local var, and finally append that to xbarv

xbarv=[]

for i in range(300):

xv = np.random.uniform(20,80,5000)

xbar = np.average(xv)

xbarv.append(xbar)

# Q4:

# Bins size should be set comparable to data difference

# Using matploit lib to represent graph

plt.hist(xbarv, bins=np.arange(min(xbarv), max(xbarv)+0.09, 0.09))

plt.title("histogram")

plt.show()

# Q5:

# merge the above code

def f_myCLT(nrSamples,sampleSize,lower,upper):

xbarv=[]

for i in range(nrSamples):

xv = np.random.uniform(lower,upper,nrSamples)

xbar = np.average(xv)

xbarv.append(xbar)

return xbarv

# Q6:

# test code

xbarv = f_myCLT(400,10000,20,80)

plt.hist(xbarv, bins=np.arange(min(xbarv), max(xbarv)+0.1, 0.1))

plt.title("histogram")

plt.show()

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