Chapter 8, Problem 94CE

This is an exercise using Excel. (a) Use =RANDBETWEEN(0,99) to create 20 samples of size n = 4 by choosing two-digit random numbers between 00 and 99. (b) For each sample, calculate the mean. (c) Make a histogram of the 80 individual x-values using bins 10 units wide (i.e., 0, 10, 20, . . . , 100). Describe the shape of the histogram. (d) Make a histogram of your 20 sample means using bins 10 units wide. (e) Discuss the histogram shape. Does the Central Limit Theorem seem to be working? (f) Find the mean of your 20 sample means. Was it what you would expect? Explain. (g) Find the standard deviation of your 20 sample means. Was it what you would expect?

To determine

Create 20 samples by choosing two digit random numbers between 00 and 99 of size $n=4$.

Answer to Problem 94CE

The random numbers are given below:

Explanation of Solution

Calculation:

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

Software procedure:

Step-by-step procedure to find the standard deviation using the EXCEL software is given below:

• • Open an EXCEL file.
• • In A1, enter the formula “=RANDBETWEEN(0,99)”.
• • Click Enter.
• • Drag the cells to bottom 4 rows and 20 columns to the right.

To determine

Calculate the mean for each sample.

Answer to Problem 94CE

The mean for each sample is given below:

Explanation of Solution

Calculation:

Mean:

The arithmetic mean(also called the average)is the most commonly used measure of central tendency. It is calculated by summing the observed numerical values of a variable in a set of data and then dividing the total by the number of observations involved.

Software procedure:

Step-by-step procedure to find the mean for each sample using the EXCEL software is given below:

• • Open an EXCEL file.
• • In A1, enter the formula “=RANDBETWEEN(0,99)”.
• • Click Enter.
• • Drag the cells to bottom 4 rows and 20 columns to the right.
• • In A5, enter the formula “=AVERAGE(A1:A4)”.
• • Click Enter.
• • Drag the cells for 20 columns to the right.

To determine

Make a histogram of 80 individual x-values using bins of 10 units wide and describe the shape of the histogram.

Answer to Problem 94CE

The histogram of 80 individual x-values using 10 units wide is given below:

Explanation of Solution

Calculation

The values are ranging from 0 to 99. The way to arrange the data in a group is, the first class takes values from0-10, and the second class takes 10-20 and so on.

Now, count the frequency (number of values) in each bin that is, 8is in bin 0-10 because 8 values occur in this interval. Similarly for the remaining bins frequencies are obtained.

The frequency table is as follows,

Class	Frequency
0-10	8
10-20	11
20-30	4
30-40	10
40-50	7
50-60	10
60-70	9
70-80	8
80-90	6
90-100	7
Total	80

Histogram:

It is bar graph for a quantitative data where the bars have a particular order and specific meaning for the bin widths.

Software procedure:

Step-by-step software procedure to obtain histogram using EXCEL is as follows:

• • Open an EXCEL file.
• • In column A, enter the column of class and in column B enter the column of Frequency.
• • Select the data that is to be displayed.
• • Select Insert > Column
• • Choose Clustered column under 2-D column.
• • Double click on the bars and reduce the Gap Width to 0
• • Click on the chart > select Layout from the Chart Tools.
• • Select Chart Title>Above Chart.
• • Enterin the dialog box.
• • Select Axis Title>Primary Horizontal Axis Title > Title Below Axis.
• • EnterClass in the dialog box.
• • Select Axis Title>Primary Vertical Axis Title > Rotated Title.
• • Enter Frequency in the dialog box.

Symmetric distribution:

The distribution is said to be symmetric when the left half is the mirror image of the right half.

Left skewed distribution:

The distribution is said to be left skewed, if most of the observations are spread out on the left side. That is, the left tail is more elongated than the right tail.

Right skewed distribution:

The distribution is said to be left skewed, if most of the observations are spread out on the right side. That is, the right tail is more elongated than the left tail.

Here, the values are spread out almost equally on both sides. Thus, the histogram is nearly symmetric.

To determine

Make a histogram of 20 sample means using bins of 10 units wide and describe the shape of the histogram.

Answer to Problem 94CE

The histogram of 20 sample means using 10 units wide is given below:

Explanation of Solution

Calculation

Here, count the frequency (number of values) in each bin that is, 0 is in bin 0-10 because none of the values of sample means occur in this interval. Similarly for the remaining bins frequencies are obtained.

The frequency table is as follows,

Class	Frequency
0-10	0
10-20	0
20-30	2
30-40	3
40-50	6
50-60	6
60-70	2
70-80	1
80-90	0
90-100	0
Total	20

Software procedure:

Step-by-step software procedure to obtain histogram using EXCEL is as follows:

• • Open an EXCEL file.
• • In column A, enter the column of class and in column B enter the column of Frequency.
• • Select the data that is to be displayed.
• • Select Insert > Column
• • Choose Clustered column under 2-D column.
• • Double click on the bars and reduce the Gap Width to 0
• • Click on the chart > select Layout from the Chart Tools.
• • Select Chart Title>Above Chart.
• • Enterin the dialog box.
• • Select Axis Title>Primary Horizontal Axis Title > Title Below Axis.
• • EnterClass in the dialog box.
• • Select Axis Title>Primary Vertical Axis Title > Rotated Title.
• • Enter Frequency in the dialog box.

To determine

Describe the shape of the histogram and explain whether the Central Limit Theorem seem to be working or not.

Answer to Problem 94CE

The shape of the histogram is symmetric. The central limit theorem seems to be working in the situation.

Explanation of Solution

For the histogram of sample means, the left half is the mirror image of the right half. Thus, the distribution of sample means is symmetric.

Central limit theorem for a mean:

If a random sample of size n is taken from a population having mean $\mu$ and standard deviation $\sigma$ then, as the sample size increases $\left(n\ge 30\right)$, the sample mean $\overline{X}$ approaches the normal distribution with mean $\mu$ and standard deviation ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$.

The three main points about the sample mean is:

• • If the population is normal, then the sample mean has a normal distribution. The mean of the distribution is $\mu$ with a standard error of $\frac{\sigma }{\sqrt{n}}$.
• • The distribution of the sample mean converges to the population mean as the sample size increases.
• • According to the theorem, even if the population is not normal, the sample means have approximately a normal distribution as the sample size becomes large.

According to the central limit theorem, distribution of the sample means converges to the normal distribution as sample size increases. Thus, the central limit theorem seems to be working in the situation.

To determine

Find the mean of the 20 sample means and explain whether it was expected one or not.

Answer to Problem 94CE

• The mean of the 20 sample means is 48.93. The obtained value of mean is close to the expected value.

Explanation of Solution

Calculation:

Software procedure:

Step-by-step procedure to find the mean using the EXCEL software is given below:

• • Open an EXCEL file.
• • Enter the sample mean values from A1 through A20.
• • In B1, enter the formula “=AVERAGE(A1:A20)”.
• • Click Enter.
• Output using EXCEL software is given below:

From the EXCEL output, the mean value is 48.93.

• Since the sample size is large the mean of the 20 sample means is expected to have the population mean. Here, random numbers are generated from 0 to 99, by uniform distribution the mean value is 49.5$\left(=\frac{b+a}{2}\right)$. Thus, the obtained value of mean is close to the expected value.

To determine

Find the standard deviation of the 20 sample means and explain whether it was expected one or not.

Answer to Problem 94CE

• The standard deviation of the 20 sample means is 12.41. The obtained value of mean is close to the expected value.

Explanation of Solution

Calculation:

Software procedure:

Step-by-step procedure to find the mean using the EXCEL software is given below:

• • Open an EXCEL file.
• • Enter the sample mean values from A1 through A20.
• • In B1, enter the formula “=STDEV(A1:A20)”.
• • Click Enter.
• Output using EXCEL software is given below:

From the EXCEL output, the standard deviation is 12.41.

• Since the sample size is large the standard deviation of the 20 sample means is expected to have thestandard deviation ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$. Here, random numbers are generated from 0 to 99, by uniform distribution the standard deviation value of population is 28.57$\left(=\frac{b-a}{\sqrt{12}}\right)$. The standard deviation of sample means is $14.285\left(=\frac{28.57}{\sqrt{4}}\right)$.
• Thus, the obtained standard deviation is close to the expected value.