Chapter 12.2, Problem 14E

How big is your family? The General Social Survey asked a sample of 2780 adults how many children they had, and also how many siblings (brothers and sisters) they had. The results are summarized in the following contingency table.

1. Compute the expected frequencies.
2. Compute the value of the test statistic.
3. How many degrees of freedom are there?
4. Test the hypothesis of independence. Using the $\alpha =0.05$ level of significance. What do you conclude?

To determine

To compute : the expected frequencies.

Answer to Problem 14E

The following table presents the expected frequencies. All the expected frequencies are at least 5.

	Number of children
Siblings	0	1	2	3	4	More Than 4
0	39.626	24.501	37.761	23.776	11.396	6.941
1	152.174	94.09	145.013	91.305	43.763	26.655
2	153	94.6	145.8	91.8	44	26.8
3	129.610	80.138	123.510	77.766	37.273	22.703
4	93.836	58.019	89.421	56.302	26.986	16.437
More Than 4	196.754	121.653	187.495	118.052	56.583	34.464

Explanation of Solution

Given information : The contingency table showing results of survey

	Number of children
Siblings	0	1	2	3	4	More Than 4
0	60	19	34	15	10	6
1	189	97	163	63	29	12
2	179	98	137	81	41	20
3	124	85	132	83	31	16
4	85	53	88	61	32	22
More Than 4	128	121	175	156	77	58

Formula Used:

  $\boxed{E=\frac{R\times C}{G}}$ , where E is expected frequency, R is row total, C is column total and G is grand total for a cell

Calculation:

Row totals reported	Column totals reported
Adults having … 0 sibling = 60+19+34+15+10+6=144 1 sibling = 189+97+163+63+29+12=553 2 siblings = 179+98+137+81+41+20=556 3 siblings = 124+85+132+83+31+16=471 4 siblings = 85+53+88+61+32+22=341 More Than 4 = 128+121+175+156+77+58=715	Adults having 0 children = 60+189+179+124+85+128 = 765 1 children = 19+97+98+85+53+121= 473 2 children = 34+163+137+132+88+175= 729 3 children = 15+63+81+83+61+156 = 459 4 children = 10+29+41+31+32+77 = 220 5 children = 6+12+20+16+22+58= 134
Grand total = 60+19+34+15+10+6+189+97+163+63+29+12+179+98+137+81+41+20+124+85+132+83+31 +16+85+53+88+61+32+22+128+121+175+156+77+58=2780

From the results we have the below table:

	Number of children
Siblings	0	1	2	3	4	More Than 4	Row Total
0	60	19	34	15	10	6	144
1	189	97	163	63	29	12	553
2	179	98	137	81	41	20	556
3	124	85	132	83	31	16	471
4	85	53	88	61	32	22	341
More Than 4	128	121	175	156	77	58	715
Column Total	765	473	729	459	220	134	2780

Finding the expected frequency for the cell corresponding to:	The expected frequency
Adults having 0 siblings and 0 children The row total is 144, the column total is 765, and the grand total is 2780.	$\frac{144\cdot 765}{2780}\approx 39.626$
Adults having 0 siblings and 1 children The row total is 144, the column total is 473, and the grand total is 2780.	$\frac{144\cdot 473}{2780}\approx 24.501$
Adults having 0 siblings and 2 children The row total is 144, the column total is 729, and the grand total is 2780.	$\frac{144\cdot 729}{2780}\approx 37.761$
Adults having 0 siblings and 3 children The row total is 144, the column total is 459, and the grand total is 2780.	$\frac{144\cdot 459}{2780}\approx 23.776$
Adults having 0 siblings and 4 children The row total is 144, the column total is 220, and the grand total is 2780.	$\frac{144\cdot 220}{2780}\approx 11.396$
Adults having 0 siblings and more than 4 children The row total is 144, the column total is 134, and the grand total is 2780.	$\frac{144\cdot 134}{2780}\approx 6.941$
Adults having 1 siblings and 0 children The row total is 553, the column total is 765, and the grand total is 2780.	$\frac{553\cdot 765}{2780}\approx 152.174$

Finding the expected frequency for the cell corresponding to:	The expected frequency
Adults having 1 siblings and 1 children The row total is 553, the column total is 473, and the grand total is 2780.	$\frac{553\cdot 473}{2780}\approx 94.090$
Adults having 1 siblings and 2 children The row total is 553, the column total is 729, and the grand total is 2780.	$\frac{553\cdot 729}{2780}\approx 145.013$
Adults having 1 siblings and 3 children The row total is 553, the column total is 459, and the grand total is 2780.	$\frac{553\cdot 459}{2780}\approx 91.305$
Adults having 1 siblings and 4 children The row total is 553, the column total is 220, and the grand total is 2780.	$\frac{553\cdot 220}{2780}\approx 43.763$
Adults having 1 siblings and more than 4 children The row total is 553, the column total is 134, and the grand total is 2780.	$\frac{553\cdot 134}{2780}\approx 26.655$
Adults having 2 siblings and 0 children The row total is 556, the column total is 765, and the grand total is 2780.	$\frac{556\cdot 765}{2780}=153$
Adults having 2 siblings and 1 children The row total is 556, the column total is 473, and the grand total is 2780.	$\frac{556\cdot 473}{2780}=94.6$
Adults having 2 siblings and 2 children The row total is 556, the column total is 729, and the grand total is 2780.	$\frac{556\cdot 729}{2780}=145.8$
Adults having 2 siblings and 3 children The row total is 556, the column total is 459, and the grand total is 2780.	$\frac{556\cdot 459}{2780}=91.8$
Adults having 2 siblings and 4 children The row total is 556, the column total is 220, and the grand total is 2780.	$\frac{556\cdot 220}{2780}=44$
Adults having 2 siblings and more than 4 children The row total is 556, the column total is 134, and the grand total is 2780.	$\frac{556\cdot 134}{2780}=26.8$
Adults having 3 siblings and 0 children The row total is 471, the column total is 765, and the grand total is 2780.	$\frac{471\cdot 765}{2780}=129.610$
Adults having 3 siblings and 1 children The row total is 471, the column total is 473, and the grand total is 2780.	$\frac{471\cdot 473}{2780}=80.138$
Adults having 3 siblings and 2 children The row total is 471, the column total is 729, and the grand total is 2780.	$\frac{471\cdot 729}{2780}=123.510$
Adults having 3 siblings and 3 children The row total is 471, the column total is 459, and the grand total is 2780.	$\frac{471\cdot 459}{2780}=77.766$
Adults having 3 siblings and 4 children The row total is 471, the column total is 220, and the grand total is 2780.	$\frac{471\cdot 220}{2780}=37.273$
Adults having 3 siblings and more than 4 children The row total is 471, the column total is 134, and the grand total is 2780.	$\frac{471\cdot 134}{2780}=22.703$
Adults having 4 siblings and 0 children The row total is 341, the column total is 765, and the grand total is 2780.	$\frac{341\cdot 765}{2780}=93.836$
Adults having 4 siblings and 1 children The row total is 341, the column total is 473, and the grand total is 2780.	$\frac{341\cdot 473}{2780}=58.019$

Finding the expected frequency for the cell corresponding to:	The expected frequency
Adults having 4 siblings and 2 children The row total is 341, the column total is 729, and the grand total is 2780.	$\frac{341\cdot 729}{2780}=89.421$
Adults having 4 siblings and 3 children The row total is 341, the column total is 459, and the grand total is 2780.	$\frac{341\cdot 459}{2780}=56.302$
Adults having 4 siblings and 4 children The row total is 341, the column total is 220, and the grand total is 2780.	$\frac{341\cdot 220}{2780}=26.986$
Adults having 4 siblings and more than 4 children The row total is 341, the column total is 134, and the grand total is 2780.	$\frac{341\cdot 134}{2780}=16.437$
Adults having more than 4 siblings and 0 children The row total is 715, the column total is 765, and the grand total is 2780.	$\frac{715\cdot 765}{2780}=196.754$
Ad Adults having more than 4 siblings and 1 children The row total is 715, the column total is 473, and the grand total is 2780.	$\frac{715\cdot 473}{2780}=121.653$
Adults having more than 4 siblings and 2 children The row total is 715, the column total is 729, and the grand total is 2780.	$\frac{715\cdot 729}{2780}=187.495$
Adults having more than 4 siblings and 3 children The row total is 715, the column total is 459, and the grand total is 2780.	$\frac{715\cdot 459}{2780}=118.052$
Adults having more than 4 siblings and 4 children The row total is 715, the column total is 220, and the grand total is 2780.	$\frac{715\cdot 220}{2780}=56.583$
Adults having more than 4 siblings and more than 4 children The row total is 715, the column total is 134, and the grand total is 2780.	$\frac{715\cdot 134}{2780}=34.464$

The following table presents the expected frequencies. All the expected frequencies are at least 5.

	Number of children
Siblings	0	1	2	3	4	More Than 4
0	39.626	24.501	37.761	23.776	11.396	6.941
1	152.174	94.09	145.013	91.305	43.763	26.655
2	153	94.6	145.8	91.8	44	26.8
3	129.610	80.138	123.510	77.766	37.273	22.703
4	93.836	58.019	89.421	56.302	26.986	16.437
More Than 4	196.754	121.653	187.495	118.052	56.583	34.464

To determine

To compute : chi-square statistic.

Answer to Problem 14E

  $\boxed{{\chi }^2=126.444}$

Explanation of Solution

Given information : The contingency table showing results of survey

	Number of children
Siblings	0	1	2	3	4	More Than 4
0	60	19	34	15	10	6
1	189	97	163	63	29	12
2	179	98	137	81	41	20
3	124	85	132	83	31	16
4	85	53	88	61	32	22
More Than 4	128	121	175	156	77	58

Formula Used:

The chi-square test statistic is ${\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}$ where O is observed frequency and E is expected frequency.

Calculation:

From the results of previous part we have the below table:

  

Finding the value of the chi-square corresponding to:	$\frac{{\left(O-E\right)}^2}{E}$
Adults having 0 siblings and 0 children The observed frequency is 60 and Expected frequency is 39.626	$\frac{{\left(60-39.626\right)}^2}{39.626}\approx 10.475$
Adults having 0 siblings and 1 children The observed frequency is 19 and Expected frequency is 24.501	$\frac{{\left(19-24.501\right)}^2}{24.501}\approx 1.235$
Adults having 0 siblings and 2 children The observed frequency is 34 and Expected frequency is 37.761	$\frac{{\left(34-37.761\right)}^2}{37.761}\approx 0.375$
Adults having 0 siblings and 3 children The observed frequency is 15 and Expected frequency is 23.776	$\frac{{\left(15-23.776\right)}^2}{23.776}\approx 3.239$
Adults having 0 siblings and 4 children The observed frequency is 10 and Expected frequency is 11.396	$\frac{{\left(10-11.396\right)}^2}{11.396}\approx 0.171$
Adults having 0 siblings and more than 4 children The observed frequency is 6 and Expected frequency is 6.941	$\frac{{\left(6-6.941\right)}^2}{6.941}\approx 0.128$
Adults having 1 siblings and 0 children The observed frequency is 189 and Expected frequency is 152.174	$\frac{{\left(189-152.174\right)}^2}{152.174}\approx 8.912$
Adults having 1 siblings and 1 children The observed frequency is 97 and Expected frequency is 94.09	$\frac{{\left(97-94.09\right)}^2}{94.09}\approx 0.09$
Adults having 1 siblings and 2 children The observed frequency is 163 and Expected frequency is 145.013	$\frac{{\left(163-145.013\right)}^2}{145.013}\approx 2.231$
Adults having 1 siblings and 3 children The observed frequency is 63 and Expected frequency is 91.305	$\frac{{\left(63-91.305\right)}^2}{91.305}\approx 8.775$
Adults having 1 siblings and 4 children The observed frequency is 29 and Expected frequency is 43.763	$\frac{{\left(29-43.463\right)}^2}{43.463}\approx 4.813$
Adults having 1 siblings and more than 4 children The observed frequency is 12 and Expected frequency is 26.655	$\frac{{\left(12-26.655\right)}^2}{26.655}\approx 8.057$
Adults having 2 siblings and 0 children The observed frequency is 179 and Expected frequency is 153	$\frac{{\left(179-153\right)}^2}{153}\approx 4.418$
Adults having 2 siblings and 1 children The observed frequency is 98 and Expected frequency is 94.6	$\frac{{\left(98-94.6\right)}^2}{94.6}\approx 0.122$
Adults having 2 siblings and 2 children The observed frequency is 137 and Expected frequency is 145.8	$\frac{{\left(137-145.8\right)}^2}{145.8}\approx 0.531$
Adults having 2 siblings and 3 children The observed frequency is 81 and Expected frequency is 91.8	$\frac{{\left(81-91.8\right)}^2}{91.8}\approx 1.271$

Finding the value of the chi-square corresponding to:	$\frac{{\left(O-E\right)}^2}{E}$
Adults having 2 siblings and 4 children The observed frequency is 41 and Expected frequency is 44	$\frac{{\left(41-44\right)}^2}{44}\approx 0.205$
Adults having 2 siblings and more than 4 children The observed frequency is 20 and Expected frequency is 26.8	$\frac{{\left(20-26.8\right)}^2}{26.8}\approx 1.725$
Adults having 3 siblings and 0 children The observed frequency is 124 and Expected frequency is 129.610	$\frac{{\left(124-129.610\right)}^2}{129.610}\approx 0.243$
Adults having 3 siblings and 1 children The observed frequency is 85 and Expected frequency is 80.138	$\frac{{\left(85-80.138\right)}^2}{80.138}\approx 0.295$
Adults having 3 siblings and 2 children The observed frequency is 132 and Expected frequency is 123.510	$\frac{{\left(132-123.510\right)}^2}{123.510}\approx 0.584$
Adults having 3 siblings and 3 children The observed frequency is 83 and Expected frequency is 77.766	$\frac{{\left(83-77.766\right)}^2}{77.766}\approx 0.352$
Adults having 3 siblings and 4 children The observed frequency is 31 and Expected frequency is 37.273	$\frac{{\left(31-37.273\right)}^2}{37.273}\approx 1.056$
Adults having 3 siblings and more than 4 children The observed frequency is 16 and Expected frequency is 22.703	$\frac{{\left(16-22.703\right)}^2}{22.703}\approx 1.979$
Adults having 4 siblings and 0 children The observed frequency is 85 and Expected frequency is 93.836	$\frac{{\left(85-93.836\right)}^2}{93.836}\approx 0.832$
Adults having 4 siblings and 1 children The observed frequency is 53 and Expected frequency is 58.019	$\frac{{\left(53-58.019\right)}^2}{58.019}\approx 0.434$
Adults having 4 siblings and 2 children The observed frequency is 88 and Expected frequency is 89.421	$\frac{{\left(88-89.421\right)}^2}{89.421}\approx 0.023$
Adults having 4 siblings and 3 children The observed frequency is 61 and Expected frequency is 56.302	$\frac{{\left(61-56.302\right)}^2}{56.302}\approx 0.392$
Adults having 4 siblings and 4 children The observed frequency is 32 and Expected frequency is 26.986	$\frac{{\left(32-26.986\right)}^2}{26.986}\approx 0.932$
Adults having 4 siblings and more than 4 children The observed frequency is 22 and Expected frequency is 16.437	$\frac{{\left(22-16.437\right)}^2}{16.437}\approx 1.883$
Adults having more than 4 siblings and 0 children The observed frequency is 128 and Expected frequency is 196.754	$\frac{{\left(128-196.754\right)}^2}{196.754}\approx 24.025$
Ad Adults having more than 4 siblings and 1 children The observed frequency is 121 and Expected frequency is 121.653	$\frac{{\left(121-121.653\right)}^2}{121.653}\approx 0.004$

Finding the value of the chi-square corresponding to:	$\frac{{\left(O-E\right)}^2}{E}$
Adults having more than 4 siblings and 2 children The observed frequency is 175 and Expected frequency is 187.495	$\frac{{\left(175-187.495\right)}^2}{187.495}\approx 0.833$
Adults having more than 4 siblings and 3 children The observed frequency is 156 and Expected frequency is 118.052	$\frac{{\left(156-118.052\right)}^2}{118.052}\approx 12.198$
Adults having more than 4 siblings and 4 children The observed frequency is 77 and Expected frequency is 56.583	$\frac{{\left(77-56.583\right)}^2}{56.583}\approx 7.367$
Adults having more than 4 siblings and more than 4 children The observed frequency is 58 and Expected frequency is 34.464	$\frac{{\left(58-34.464\right)}^2}{34.464}\approx 16.073$

To compute the test statistics, we use the observed frequencies and expected frequency:

  $\begin{array}{l}\begin{array}{l}\\ \\ {\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}\\ \\ \end{array}\}=\begin{array}{c}\begin{array}{l}10.475+1.235+0.375+3.239+0.171+0.128\\ +8.912+0.09+2.231+8.775+4.813+8.057\\ +4.418+0.122+0.531+1.271+0.205+1.725\\ +0.243+0.295+0.584+0.352+1.056+1.979\\ +0.832+0.434+0.023+0.392+0.932+1.883\\ +24.025+0.004+0.833+12.198+7.367+16.073\end{array}\end{array}\\ {\chi }^2\approx 126.444\end{array}$

To determine

To find : the number of degrees of freedom.

Answer to Problem 14E

The degree of freedom is $\boxed{25}$

Explanation of Solution

Given information : The contingency table showing results of survey

	Number of children
Siblings	0	1	2	3	4	More Than 4
0	60	19	34	15	10	6
1	189	97	163	63	29	12
2	179	98	137	81	41	20
3	124	85	132	83	31	16
4	85	53	88	61	32	22
More Than 4	128	121	175	156	77	58

Calculation:

A contingency table relates two qualitative variables. The row variables has on category for each row of the table. The column variable has one category for each column of the table.

$Thenumberofdegreeoffreedom=\left(r-1\right)\left(c-1\right)$ ; Here r represents the number of rows and c represents the number of columns. Given $r=6$ , $c=6$ so the number of degree of freedom is $\left(6-1\right)\left(6-1\right)=\left(5\right)\left(5\right)=25$

To determine

To perform : a test of independence, using the $\alpha =0.05$ level of significance

Answer to Problem 14E

The test statistics ${\chi }^2=126.444$ is greater than the critical value 37.652, so we reject $H_0$ . We conclude that the number of siblings and the number of children are not independent.

Explanation of Solution

Given information : The contingency table showing results of survey

	Number of children
Siblings	0	1	2	3	4	More Than 4
0	60	19	34	15	10	6
1	189	97	163	63	29	12
2	179	98	137	81	41	20
3	124	85	132	83	31	16
4	85	53	88	61	32	22
More Than 4	128	121	175	156	77	58

Concept involved:

Find the critical value from Table A.4, using $\left(r-1\right)\left(c-1\right)$ degrees of freedom, where r is the number of rows and c is the number of columns. If chi-square statistics ${\chi }^2$ is greater than or equal to the critical value, reject $H_0$ . Otherwise, do not reject $H_0$ .

Calculation:

Degrees of freedom	Table A.4 Critical Values for the chi-square Distribution
	0.995	0.99	0.975	0.95	0.90	0.10	0.05	0.025	0.01	0.005
25	10.520	11.524	13.120	14.611	16.473	34.382	37.652	40.646	44.314	46.928