A variable is said to be uniformly distributed or to have a uniform distribution with parameters \(a\) and \(b\) if its distribution has the shape of the horizontal line segment with equation \(y=\frac{1}{(b-a)}\) for \(a<x<b\). The mean and standard deviation of such a variable are \(\frac{(a+b)}{2}\) and \(\frac{(b-a)}{\sqrt{12}}\) respectively. The basic random number generator on a computer or calculator which returns a number between \(0\) and \(1\), simulates a variable having a uniform distribution with parameters \(0\) and \(1\).

a. Sketch the distribution of a uniformly distributed variable with parameters \(0\) and \(1\). Observe from your sketch that such a variable is far from being normally distributed.

b. Use the technology of your choice to simulate \(2000\) samples of two random numbers between \(0\) and \(1\).

c. Find the sample mean of each of the \(2000\) samples obtained in part (b).

d. Determine the mean and standard deviation of the \(2000\) sample means.

e. Theoretically what are the mean and the standard deviation of all possible sample means for samples of size \(2\)? Compare your answers to those you obtained in part (d).

f. Obtained a histogram of the \(2000\) sample means. Is the histogram bell shaped? Would you expect it to be?

g. Repeat parts (b)-(f) for a sample size of \(35\).

The value of variable is

\(a=0\)

\(b=1\)

\(n=2\)

Calculate the mean and standard deviation using given relation

\(mean=\frac{a+b}{2}=0.5\)

\(\sigma=\frac{(b-a)}{\sqrt{12}}=0.2887\)

The compute the value of the function

\(y=\frac{1}{b-a}=1\)

So, we can see that the function value is 1 then sketch a graph using MATLAB.

Program:

Query:

• First, we have defined given variables.
• Then calculate the mean and standard deviation.
• Then calculate the value of the function.

• Sketch a graph.

So, according to the question I have used MATLAB to generate the samples.

We use the function \(“norminv (rand (1000,4), 49.34, 10.14)”\).

After using the above function in MATLAB, we will get the \(1000\) sample for \(4\) dancers.

Program:

Query:

• Then using function “norminv” to find out the samples for dancers.

So, according to the question I have used MATLAB to generate the samples.

We use the function \(“norminv (rand (1000,4), 49.34, 10.14)”\).

Then calculate the mean

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

After solving the equation, we will get

\(\mu =[49.3615, 49.1953, 48.7853,49.1773]\)

Program:

Query:

• First, we have defined the number of samples.
• Then using function “norminv” to find out the samples for dancers.

• Then calculate the mean using function “mean”.

So, according to the question I have used MATLAB to generate the samples.

We use the function \(“norminv (rand (1000,4), 49.34, 10.14)”\).

Then calculate the mean

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

After solving the equation, we will get

\(\mu =[49.3615, 49.1953, 48.7853,49.1773]\)

Calculate the mean of means

\(\mu_{\bar{x}}=\bar{\mu}\)

and standard deviation

\(\sigma_{\bar{x}}=\frac{10.14}{\sqrt{4}}=5.07\)

Program:

Query:

• First, we have defined the number of samples.
• Then using function “norminv” to find out the samples for dancers.
• Then calculate the mean using function “mean”.
• Calculate the mean of means and standard deviation.

The mean of the sampling distribution is equal to the population mean.

\(\mu_{\bar{x}}=\mu=49.34\)

The standard deviation of the sampling distribution is equal to the population standard deviation divide by the square root of the sample size.

\(\sigma_{\bar{x}}=\frac{10.14}{\sqrt{4}}=5.07\)

So, according to the question I have used MATLAB to generate the samples.

We use the function \(“norminv (rand (1000,4), 49.34, 10.14)”\).

Then calculate the mean

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

After solving the equation, we will get

\(\mu =[49.3615, 49.1953, 48.7853,49.1773]\)

Calculate the mean of means

\(\mu_{\bar{x}}=\bar{\mu}\)

\(\mu_{\bar{x}}=49.6020\)

and standard deviation

\(\sigma_{\bar{x}}=\frac{10.14}{\sqrt{4}}=5.07\)

Program:

Query:

• First, we have defined the number of samples.
• Then using function “norminv” to find out the samples for dancers.
• Then calculate the mean using function “mean”.

• Calculate the mean of means and standard deviation.
• Then sketch a histogram of the samples.