Chapter 11.1, Problem 12E

To determine

To find: Whether there is sufficient evidence to conclude that there is a significant different in the distribution of city by race at 5% significance level.

Answer to Problem 12E

Yes, there is a significant different in the distribution of city by race at 5% significance level.

Explanation of Solution

Given:

The proportion values are 0.28, 0.24, 0.35, 0.12 and 0.01.

The table is:

212

202

270

94

22

Formula used:

The formula to compute the test statistic is

  ${\chi }^2={\displaystyle \sum \frac{{\left(O-E\right)}^2}{E}}$

Calculation:

The null and alternative hypotheses are:

  $\begin{array}{l}{H}_0:p_1=0.28,p_2=0.24,p_3=0.35,p_4=0.12,p_5=0.01\\ H_a:\text{ At least one of }{p}_i\text{ is different}\end{array}$

The calculation for test statistic is done as:

Observed value	Expected value	$\left(O-E\right)$	${\left(O-E\right)}^2$	${\left(O-E\right)}^2/E$
0.28	224	-12	144	0.6429
0.24	192	10	100	0.5208
0.35	280	-10	100	0.3571
0.12	96	-2	4	0.0417
0.01	8	14	196	24.5
				${\displaystyle \sum {\frac{\left(O-E\right)}{E}}^2=26.0625}$

The test statistic is:

  $\begin{array}{c}{\chi }^2={\displaystyle \sum {\frac{\left(O-E\right)}{E}}^2}\\ =26.0625\end{array}$

The degree of freedom is calculated as:

  $\begin{array}{c}\text{Degree of freedom =Number of categories - 1}\\ =5-1\\ =4\end{array}$

The p-value using chi-square table at 4 degree of freedom is 0.00003

The p -value is below significance level. The null hypothesis gets rejected. Thus, there is a significant different in the distribution of city by race at 5% significance level.