Chapter 6, Problem 64CE

(a)

To determine

Generate the 1,000 random numbers between 1 through 5.

Find the expected mean and standard deviation.

Answer to Problem 64CE

The mean value for the numbers between 1 and 5 is 3.

The standard deviation value for the numbers between 1 and 5 is 1.414.

Explanation of Solution

Calculation:

Generate 1,000 random numbers:

The formula to generate random numbers between two numbers by using EXCEL is RANDBETWEEN (bottom, top).

Software procedure:

Step by step procedure to obtain the 1,000 random numbers by using EXCEL software:

• • Open EXCEL.
• • In cell A1, enter the value as “=RANDBETWEEN(1,5)”.

Output using EXCEL software is given as follows:

4	5	2	4	5	2	2	2	4	5	...	5	5	3	5	1	2	4	4	2	3
3	2	4	5	5	3	3	4	5	3	…	5	5	4	5	2	4	1	5	5	2
3	4	5	2	5	5	2	4	2	1	…	1	2	4	2	4	3	1	5	3	5
5	4	5	4	5	3	2	1	2	5	…	1	3	3	1	5	4	3	2	2	5
5	4	2	1	3	2	2	3	5	4	…	4	2	5	4	3	2	4	4	4	3
4	4	3	2	4	3	5	3	4	5	…	2	3	4	2	5	1	4	1	5	3
2	5	4	5	2	4	5	4	3	4	…	3	5	3	2	2	5	1	5	5	4
4	5	1	2	5	1	2	5	5	3	…	4	1	3	4	4	3	5	1	3	3
4	4	1	1	3	5	4	4	2	2	…	1	3	1	1	1	3	3	5	1	5
2	1	1	4	1	1	3	4	1	2	…	5	5	3	5	1	2	4	4	2	3

Uniform distribution:

The random variable with a finite number of integers from a and b follows uniform distribution with parameters a and b.

The PDF of the distribution is,

$P\left(X=x\right)=\frac{1}{b-a+1}$.

The formula for mean is,

$\mu =\frac{a+b}{2}$.

The formula for standard deviation is,

$\sigma =\sqrt{\frac{{\left[\left(b-a\right)+1\right]}^2-1}{12}}$

The mean value for values between 1 and 5 is,

$\begin{array}{c}\mu =\frac{1+5}{2}\\ =\frac{6}{2}\\ =3\end{array}$

Thus, the value of mean for 1,000 random numbers between 1 and 5 is 3.

The standard deviation value for values between 1 and 5 is,

$\begin{array}{c}\sigma =\sqrt{\frac{{\left[\left(5-1\right)+1\right]}^2-1}{12}}\\ =\sqrt{\frac{{\left[\left(4\right)+1\right]}^2-1}{12}}\\ =\sqrt{\frac{{\left[5\right]}^2-1}{12}}\end{array}$

   $\begin{array}{c}=\sqrt{\frac{25-1}{12}}\\ =\sqrt{\frac{24}{12}}\\ =\sqrt{2}\\ =1.414\end{array}$

Thus, the value of standard deviation value for 1,000 random numbers between 1 and 5 is 1.414.

(b)

To determine

Find the sample mean and standard deviation.

Answer to Problem 64CE

The sample mean value for the numbers between 1 and 5 is 3.013.

The sample standard deviation value for the numbers between 1 and 5 is 1.4137.

Explanation of Solution

Calculation:

Answer will vary. One of the possible answer is given below:

Sample mean:

Software procedure:

Step by step procedure to obtain the sample mean for 1,000 random numbers by using EXCEL software:

• • Enter the formula “=AVERAGE (A1:A1000).

Output using EXCEL software is given as follows:

Thus, the value of sample mean for 1,000 random numbers between 1 and 5 is 3.013.

Sample standard deviation:

Software procedure:

Step by step procedure to obtain the sample mean for 1,000 random numbers by using EXCEL software:

• • Enter the formula “=STDEVA (A1:A1000).

Output using EXCEL software is given as follows:

Thus, the value of sample standard deviation for 1,000 random numbers between 1 and 5 is 1.4137.

(c)

To determine

Check the sample is consistent with uniform model.

Answer to Problem 64CE

The sample is consistent with uniform model.

Explanation of Solution

Calculation:

Answer will vary. One of the possible answer is given:

From the result of part (a): the results are drawn from the uniform distribution.

The mean value for the numbers between 1 and 5 is 3 and the standard deviation value for the numbers between 1 and 5 is 1.414.

From the result of part (b): the results are drawn from the samples.

The sample mean value for the numbers between 1 and 5 is 3.013 and the sample standard deviation value for the numbers between 1 and 5 is 1.4137.

By observing the results, in both cases the results are same. Thus, the sample is consistent with uniform model.

(d)

To determine

Give the EXCEL formulae used to generate random numbers.

Explanation of Solution

Generate the 1,000 random numbers between 1 through 5 is “=RANDBETWEEN(1,5)”.