Chapter 3, Problem 12E

(a)

To determine

To find: The people who are selected for the simple random sample.

Answer to Problem 12E

Solution: The selected people are those who are labeled as 19, 22, 39, and 50.

Explanation of Solution

Calculation:

The people are labeled as $00,01,\dots ,99$. The people are chosen using the random number table provided in the book. The two-digit random numbers are chosen from the random number table starting from the 101th row and the 1st column of the table. The random number is selected from 00 to 99. If the same random number is repeated after appearing for the first time, it will be rejected, and if the selected number is more than 99, then that number is also rejected. The selected random numbers are 19, 22, 39, and 50.

The people selected are the people who are numbered with the same number of the selected random numbers.

To determine

To find: The proportion, $\widehat{p}$.

Answer to Problem 12E

Solution: The required proportion is obtained as 0.5.

Explanation of Solution

Calculation:

According to the question, the colored circle indicates the persons who are opposing the constitutional amendment and the white circle represents the people who are in favor of the amendment. The selected persons are 19, 22, 39, and 50. Among them, 19 and 50 are the colored circles and 22 and 39 are the white circles.

Therefore, the proportion $\widehat{p}$ of the persons who are in favor of the amendment in the sample is calculated as follows:

$\begin{array}{c}\widehat{p}=\frac{2}{4}\\ =0.5\end{array}$

(b)

To determine

To find: The people who are selected for the simple random sample.

Answer to Problem 12E

Solution: The selected people are those people who are labeled with the same number of selected random number and the samples are shown in the below table.

Sample	Selected label
Sample 1	45, 46, 77, 17
Sample 2	52, 71, 13, 88
Sample 3	95, 59, 29, 40
Sample 4	68, 41, 73, 50
Sample 5	82, 73, 95, 78
Sample 6	60, 94, 07, 20
Sample 7	36, 00, 91, 93
Sample 8	38, 44,84, 87
Sample 9	73, 67, 64, 71

Explanation of Solution

Calculation:

The people are labeled as $00,01,\dots ,99$. The people are chosen using the random number table provided in the book. The two-digit random numbers are chosen from the random number table starting from the 102th row and the 1st column of the table. The random number is selected from 00 to 99. If the same random number is repeated after appearing for the first time, it will be rejected, and if the selected number is more than 99, then that number is also rejected. The selected random numbers are 73, 67, 64, and 71.

The people selected are the people who are numbered with the same number of the selected random numbers.

Similarly, the other eight samples of size 4 are selected and the samples are shown in the below table.

Sample	Selected label
Sample 1	45, 46, 77, 17
Sample 2	52, 71, 13, 88
Sample 3	95, 59, 29, 40
Sample 4	68, 41, 73, 50
Sample 5	82, 73, 95, 78
Sample 6	60, 94, 07, 20
Sample 7	36, 00, 91, 93
Sample 8	38, 44, 84, 87

Here, the people selected in each set of sample are the people who are numbered with the same number of the selected random numbers.

To determine

To find: The proportion, $\widehat{p}$.

Answer to Problem 12E

Solution: The required proportions for the 10 samples are shown below.

Sample	Labeled value	Proportion
Sample 1	45, 46, 77, 17	0.75
Sample 2	52, 71, 13, 88	1
Sample 3	95, 59, 29, 40	0.75
Sample 4	68, 41, 73, 50	0.5
Sample 5	82, 73, 95, 78	0.75
Sample 6	60, 94, 07, 20	0
Sample 7	36, 00, 91, 93	0.75
Sample 8	38, 44,84, 87	0.5
Sample 9	73, 67, 64, 71	0.5
Sample 10	19, 22, 39, 50	0.5

Explanation of Solution

Calculation:

The calculation of the sample proportion, $\widehat{p}$, is shown in the below table for all the 10 samples.

Sample	Labeled value	Colored circle	White circle	Proportion
Sample 1	45, 46, 77, 17	77	45, 46, 17	$\begin{array}{c}\widehat{p}=\frac{3}{4}\\ =0.75\end{array}$
Sample 2	52, 71, 13, 88	88	52, 71, 13	$\begin{array}{c}\widehat{p}=\frac{4}{4}\\ =1\end{array}$
Sample 3	95, 59, 29, 40		29, 40, 95, 59	$\begin{array}{c}\widehat{p}=\frac{3}{4}\\ =0.75\end{array}$
Sample 4	68, 41, 73, 50	73, 50	68, 41	$\begin{array}{c}\widehat{p}=\frac{2}{4}\\ =0.5\end{array}$
Sample 5	82, 73, 95, 78	73,	82, 95, 78	$\begin{array}{c}\widehat{p}=\frac{3}{4}\\ =0.75\end{array}$
Sample 6	60, 94, 07, 20	60, 94, 07, 20		$\begin{array}{c}\widehat{p}=\frac{0}{4}\\ =0\end{array}$
Sample 7	36, 00, 91, 93	91	00, 36, 93	$\begin{array}{c}\widehat{p}=\frac{3}{4}\\ =0.75\end{array}$
Sample 8	38, 44,84, 87	38, 87	44, 84	$\begin{array}{c}\widehat{p}=\frac{2}{4}\\ =0.5\end{array}$
Sample 9	73, 67, 64, 71	73, 64	67, 71	$\begin{array}{c}\widehat{p}=\frac{2}{4}\\ =0.5\end{array}$
Sample 10	19, 22, 39, 50	22, 39	19, 50	$\begin{array}{c}\widehat{p}=\frac{2}{4}\\ =0.5\end{array}$

(c)

To determine

To graph: The histogram.

Answer to Problem 12E

Solution: The required histogram is shown below.

Explanation of Solution

The 10 sample proportions are marked in the horizontal axis and the number of occurrence is marked in the vertical axis. The horizontal axis has five points such as 0, 0.25, 0.5, 0.75, and 1. The vertical axis shows 10 points such as $0,1,\dots ,10$. The bar is drawn over the each point of the horizontal axis and the height of the bar is equal to the number of the occurrence of the particular value in 10 samples. Therefore, the histogram is obtained as follows:

Interpretation: From the hisogram it is clear that none of the sample has sample proportion of 0.25.

(d)

To determine

The number of sample proportion that estimates the population proportion as $p=0.5$.

Answer to Problem 12E

Solution: There are 4 samples that provide the proportion as 0.5.

Explanation of Solution

The sample proportion obtained in the previous part shows that there are 4 samples whose sample proportion of the persons who are in favor of the constitutional amendment is 0.5.

To determine

Whether the value 0.5 is in the center of the sample values.

Answer to Problem 12E

Solution: The original value 0.5 is in the center for the sample proportion values.

Explanation of Solution

From the histogram obtained in the previous part, it can be said that the original value of the population proportion 0.5 is in the center for the sample values also.

To determine

To explain: The reason that the population proportion value will be in the center for the large samples.

Answer to Problem 12E

Solution: The value will be centered in 0.5 for large samples because the large samples have less variability than the small samples.

Explanation of Solution

Though the estimated proportion varies from sample to sample, for a large sample, the value will not differ greatly. As the large sample has less variability compared to the small sample, it can be said that the true value of the population proportion will be centered for the sample proportion value also.