Chapter 3, Problem 13E

(a)

Section 1:

To determine

The 20 simple random samples of size 5 and record the number of females in each sample.

Answer to Problem 13E

Solution: The partial output of the 20 simple random samples is shown below:

The number of females in each sample is shown below.

Samples	Number of females	Samples	Number of females
Sample 1	2	Sample 11	1
Sample 2	1	Sample 12	2
Sample 3	2	Sample 13	4
Sample 4	3	Sample 14	2
Sample 5	1	Sample 15	3
Sample 6	2	Sample 16	2
Sample 7	3	Sample 17	0
Sample 8	1	Sample 18	2
Sample 9	1	Sample 19	1
Sample 10	2	Sample 20	2

Explanation of Solution

To obtain the 20 simple random samples of size 5, below steps are followed in Excel.

Step 1: Write the name and the gender of the club members in the Excel spreadsheet. The snapshot is shown below:

Step 2: Label each of the members using the numbers $1,2,\cdots ,25$. The snapshot is shown below:

Step 3: Use the formula $=\text{RANDBETWEN}\left(\text{Bottom, Top}\right)$ to generate a single set of sample of size 5. Specify the “bottom” value as 1 and “top” value as 25 because there are 25 members. Discard the number if the same number is repeated after appearing once. Repeat the process to generate 20 set of random sample of size 5. The partial snapshot of the chosen random numbers is shown below:

To find the number of female in each sample, the label of each female member is matched with the obtained random number of each set of sample and then the number of female is counted. The number of females obtained in each set of samples are shown in the below table.

Samples	Number of females	Samples	Number of females
Sample 1	2	Sample 11	1
Sample 2	1	Sample 12	2
Sample 3	2	Sample 13	4
Sample 4	3	Sample 14	2
Sample 5	1	Sample 15	3
Sample 6	2	Sample 16	2
Sample 7	3	Sample 17	0
Sample 8	1	Sample 18	2
Sample 9	1	Sample 19	1
Sample 10	2	Sample 20	2

To determine

To graph: The histogram for the number of females in each set of sample.

Explanation of Solution

Graph: To obtain the histogram for the obtained result in previous part, Excel is used. The below steps are followed to obtain the required histogram.

Step 1: The number of females varies from 0 to 4. The number of samples are calculated corresponding to each number of females. The number of samples with the same number of female members is clustered. The snapshot of the obtained table is shown below:

Number of female members	Number of samples
0	1
1	6
2	9
3	3
4	1

Step 2: Select the data set and go to insert and select the option of cluster column under the Recommended Charts. The screenshot is shown below:

Step 3: Click on OK. The diagram is obtained as:

Step 4: Click on the chart area and select the option of “Primary Horizontal” and “Primary Vertical” axis under the “Add Chart Element” to add the axis title. The screenshot is shown below:

Step 5: Click on the bars of the diagrams and reduce the gap width to zero under the “Format Data Series” tab. The screenshot is shown below:

The obtained histogram is:

Interpretation: The obtained histogram for the number of females in each set of sample follows normal distribution.

To determine

To find: The average number of females in 20 samples.

Answer to Problem 13E

Solution: The average number of female is 1.85.

Explanation of Solution

Calculation:

The average number of females can be obtained by using the formula:

$\text{Average}=\frac{\text{Sum of}\left(\text{Number of female memebers}\times \text{Number of samples}\right)}{\text{Total number of samples}}$

Hence, the average number of females is calculated as:

$\begin{array}{c}\text{Average}=\frac{\text{Sum of}\left(\text{Number of female memebers}\times \text{Number of samples}\right)}{\text{Total number of samples}}\\ =\frac{\left(0\times 1\right)+\left(1\times 6\right)+\left(2\times 9\right)+\left(3\times 3\right)+\left(4\times 1\right)}{20}\\ =1.85\end{array}$

Therefore, the average number of females in 20 samples is 1.85.

(b)

To determine

To explain: Whether the club members should be doubtful about the discrimination if any of the five tickets does not received by women.

Answer to Problem 13E

Solution: The club member should suspect the discrimination if any of the five tickets does not received by women.

Explanation of Solution

The event is going to be suspected if the chance of occurrence of that event is approximately 0.05 or less. In the provided scenario, there will be one case where the set of samples does not contain any female member. As the chance is very low, the club members can suspect about the discrimination if any of the five tickets does not received by women.