Q. 30 - The Practice of Statistics for AP

Question

Refer to Exercise 28. Do the data provide convincing evidence of a difference in the distributions of opinions about how high schools are doing among black, Hispanic, and white parents? 

(a) State appropriate null and alternative hypotheses for a significance test to help answer this question. 

(b) Calculate the expected counts. Show your work. 

(c) Calculate the chi-square statistic. Show your work. 

Step-by-Step Solution

Answer

(a) $H_0$: There is no association between the variables

$H_a$: There is an association between the variables

(b) The expected counts is 

(c) The value of ${\chi }^2=22.4261.$

Part (a) Step 1: Given information

The given data is 

Part (a) Step 2: Explanation

The null hypothesis states that there is no association between the variables: $H_0$: There is no association between the variables

The alternative hypothesis states that there is an association between the variables:

$H_a$: There is an association between the variables

Part (b) Step 1: Given information

The given data is

Part (b) Step 2: Explanation

The given data is

$n_1=202$

$n_2=202$

$$\begin{gathered} n=202+202+201 \\ =605 \end{gathered}$$

The expected count is the row total multiplied by the sample size$n_i$ divided by the total sample size n :

Part (c) Step 1: Given information

The given data is 

Part (c) : Step 2: Explanation

Observed counts

Expected counts

The chi-square statistic is the sum of squared deviations (between observed and expected counts) divided by the expected count:

$$\begin{gathered} {\chi }^2=\frac{(12-22.7041{)}^2}{22.7041}+\frac{(69-68.4463{)}^2}{68.4463}+\frac{(75-65.4413{)}^2}{65.4413}+\frac{(24-24.0397{)}^2}{24.0397}+\frac{(22-21.3686{)}^2}{21.3686} \\ +\frac{(34-22.7041{)}^2}{22.7041}+\frac{(55-68.4463{)}^2}{68.4463}+\frac{(61-65.4413{)}^2}{65.4413}+\frac{(24-24.0397{)}^2}{24.0397}+\frac{(28-21.3686{)}^2}{21.3686} \\ +\frac{(22-22.5917{)}^2}{22.5917}+\frac{(81-68.1074{)}^2}{68.1074}+\frac{(60-65.1174{)}^2}{65.1174}+\frac{(24-23.9207{)}^2}{23.9207}+\frac{(14-21.2628{)}^2}{21.2628} \\ =22.4261 \end{gathered}$$