Q. 6.48 - Elementary Statistics

Question

Refer to the simulation of human gestation periods discussed in Example \(6.4\) on page \(259\).

a. Sketch the normal curve for human gestation periods

b. Simulate \(1000\) human gestation periods.

c. Approximately what values would you expect for the sample mean and sample deviation of the \(1000\) observations? Explain your answers.

d. Obtain the sample mean and sample standard deviation of the \(1000\) observations, and compare your answers to your estimates in part (c).

e. Roughly what would you expect a histogram of the \(1000\) observation to look like? Explain your answer.

f. Obtain a histogram of the \(1000\) observations, and compare your result to your expectation in part (e). 

Step-by-Step Solution

Answer

Part a. 

Part b. 

Part c. The expected population’s mean will be \(266\) and standard \(16\).

Part d. The population’s mean will be \(\bar{x}=266.62\) and standard \(s=15.8152\)

Part e. 

Part a. Step 1. Given information

The mean and standard are given

\(\mu =266\)

\(\sigma =16\)

Part a. Step 2. Calculation

Randomly generate \(x=1000\) numbers from the normal distribution with mean \(\mu=266\) and standard deviation \(\sigma=16\).

Then Using MATLAB draw a graph.

Program:

clc

clear

close all

\(x=norminv(rand(1000,1),266,16);\)

hist(x)

\(set(gca,'linewidth',1.2,'fontsize',12)\)

axis square

Query:

• First, we have defined the \(1000\) random numbers whose mean and standard deviation is \(266\) and \(16\).
• Then using function “hist” draw a figure.

Part b. Step 1. Calculation

Randomly generate \(x=1000\) numbers from the normal distribution with mean \(\mu=266\) and standard deviation \(\sigma=16\).

Then Using MATLAB generate the matrix of \(1000\) numbers.

Program:

clc

clear

close all

\(x=norminv(rand(1000),266,16)\);

Query:

• First, we have defined the \(1000\) random numbers whose mean and standard deviation is \(266\) and \(16\).

• Then using function “hist” draw a figure.

Part c. Step 1. Given information

The number of observations is given

\(x=500\)

Part c. Step 2. Calculation

Calculate the mean:

\(\bar{x}=\frac{\sum_{i-1}^{n}x_{i}}{n}=\frac{\sum_{i-1}^{1000}x_{i}}{1000}\)

The expected mean will be

\(\bar{x}=266\)

Calculate the expected standard deviation

\(s=\sqrt{\frac{\sum_{i-1}^{n}(x_{i}-\bar{x})^{2} }{n-1}}=\sqrt{\frac{\sum_{i-1}^{1000}(x_{i}-\bar{x})^{2} }{999}}\)

After calculating we will get

\(s=16\)

The population mean will be \(266\) and standard \(16\).

Part d. Step 1. Calculation

Calculate the mean:

\(\bar{x}=\frac{\sum_{i-1}^{n}x_{i}}{n}=\frac{\sum_{i-1}^{1000}x_{i}}{1000}\)

The expected mean will be

\(\bar{x}=266.62\)

Calculate the expected standard deviation

\(s=\sqrt{\frac{\sum_{i-1}^{n}(x_{i}-\bar{x})^{2}}{n-1}}=\sqrt{\frac{\sum_{i-1}^{1000}(x_{i}-\bar{x})^{2} }{999}}\)

After calculating we will get

\(s=15.8152\)

The population’s mean will be \(\bar{x}=266.62\) and standard \(s=15.8152\).

Part e. Calculation

Randomly generate \(x=1000\) numbers from the normal distribution with mean \(\mu=266\) and standard deviation \(\sigma=16\).

Then Using MATLAB draw a graph.

Program:

clc

clear

close all

\(x=norminv(rand(1000,1),266,16);\)

hist(x)

\(set(gca,'linewidth',1.2,'fontsize',12)\)

axis square

Query:

• First, we have defined the \(1000\) random numbers whose mean and standard deviation is \(266\) and \(16\).

• Then using function “hist” draw a figure.