91Ó°ÊÓ

In this exercise, you are to perform a computer simulation to illustrate the sampling distribution of the difference between two sample means for independent samples, Key Fact $10.1$ on page $404$ .
(a) Simulate$1000$samples of size $12$from a normally distributed variable with a mean of 640 and a standard deviation of $70$. Obtain the sample mean of each of the$1000$samples.
(b) Simulate $1000$samples of size 15 from a normally distributed variable with a mean of 715 and a standard deviation of 150 . Obtain the sample mean of each of the $1000$ samples.
(c) Obtain the difference, $\overline{x_1}−\overline{x_2}$, for each of the $1000$ pairs of sample means obtained in parts (a) and (b).
(d) Obtain the mean, the standard deviation, and a histogram of the $1000$differences found in part (c).

Step by step solution

01

Part (a) Step 1:Given information

Create a random $1000$sample with a standard deviation of$70$and a sample mean of 640.

02

Explanation

Given that, The sample size ($n$), mean (role="math" localid="1653901375641" $\overline{x}$), and standard deviation ($\mathrm{σ}$) are all given.$x¯=640,\mathrm{σ}=70,n=1000,\mathrm{σ}=70,n=1000$
Calculation:
In MATLAB, generate $1000$male samples using the function "norminv" with $640$sample mean and $70$standard deviation. In MATLAB, generate 1000 male samples using the function "norminv" with $640$sample mean and 70 standard deviation.

$x=norminv\left(rand\right(1000,12),640,70)$

Following that, we will receive $1000$random samples.
Program:
clc
clear
close all
$$\begin{gathered} n=1000; \\ r=norminv\left(rand\right(n,12),640,70); \\ x=mean(r,2); \end{gathered}$$
Query:

• First, we established the number of samples.
• Then, using the function "norminv," generate $1000$random samples with a sample mean of $640$and a standard deviation of $70$.

03

Part (b) Step 1: Given information

Define a random $1000$-sample sample with a standard deviation of $150$ and a sample mean of 715.

04

Explanation

Given that, the sample size ($n$), mean (role="math" localid="1653901451323" $\overline{x}$), and standard deviation ($\mathrm{σ}$) are all given.

$\overline{x}=715,\mathrm{σ}=150,n=1000$

Calculation:
In MATLAB, generate $1000$male samples with a sample mean of $715$and a standard deviation of $150$using the function "norminv."

$x_1=norminv\left(rand\right(1000,12),715,150)$

Following that, we will obtain 1000 random samples.
Program:
clc
clear
close all
$$\begin{gathered} n=1000; \\ r=norminv\left(rand\right(n,12),715,150); \end{gathered}$$
Query:

• First, we established the number of samples.
• Then, using the function "norminv," generate $1000$random samples with a sample mean of $715$and a standard deviation of $150$.

05

Part (c) Step 1: Given information

Determine the difference between the samples from parts (a) and (b).

06

Explanation

Given that, The sample size ($n$), mean ($\overline{x}$), and standard deviation ($\mathrm{σ}$) are all given.
$$\begin{gathered} \overline{x}=640,\mathrm{σ}=70,n=1000, \\ \overline{x}=715,\mathrm{σ}=150,n=1000, \end{gathered}$$
Calculation:
In MATLAB, generate 1000 male samples using the function "norminv" with $640$sample mean and $75$standard deviation.

$x=norminv\left(rand\right(1000,12),640,70)$

Then, in MATLAB, generate $1000$male samples with a sample mean of $715$and a standard deviation of $150$using the function "norminv."

$x_1=norminv\left(rand\right(1000,12),715,150)$

After that, subtract both samples.

$a=x−x_1$

07

Program

clc
clear
close all
$$\begin{gathered} n=1000; \\ r=norminv\left(rand\right(n,12),640,70); \\ x=mean(r,2); \end{gathered}$$
$$\begin{gathered} r1=norminv\left(rand\right(n,12),715,150); \\ x1=mean(r1,2); \end{gathered}$$
$a=x-x1;$
Query:

• First, we established the number of samples.
• Then, using the function "norminv," generate $1000$random samples with a sample mean of $640$and a standard deviation of $75$.
• The sample is then generated with a mean of $715$and a standard deviation of $150$.

  Subtraction of both samples

08

Part (d) Step 1: Given information

Create a histogram of all samples from the part (c).

09

Explanation

Given that,The sample size ($n$), mean ($\overline{x}$), and standard deviation ($\mathrm{σ}$) are all given.

$$\begin{gathered} \overline{x}=640,\mathrm{σ}=70,n=1000 \\ \overline{x}=715,\mathrm{σ}=150,n=1000 \end{gathered}$$

Calculation:
In MATLAB, generate 1000 male samples using the function "norminv" with $640$sample mean and $75$standard deviation.

$x=norminv\left(rand\right(1000,12),640,70)$

Then, in MATLAB, generate male sample means of $715$and standard deviations of $150$.

$x1=norminv\left(rand\right(1000,12),715,150)$

After that, subtract both samples.

$a=x−x1$

The histogram is then generated using MATLAB.

10

Program

Clc
clear
close all
$$\begin{gathered} n=1000; \\ r=norminv(rand(n,12),640,70); \\ x=mean(r,2); \\ r1=norminv\left(rand\right(n,12),715,150); \\ x1=mean(r1,2); \\ a=x−x1; \end{gathered}$$
Query:

• First, we established the number of samples.
• The random samples are then generated.
• The histogram is then generated.

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