Question

In a recent national poll, people were asked the following question: "In your opinion, how important is it to improve the nation's inner-city schools?" The responses of city residents who do not have school-age children were compared to the national responses. A chi square test was used to analyze the data in order to determine whether there is a difference in responses between those who live in cities and do not have school-age children and the national responses. The results of the study are displayed in the following table. The analysis revealed a chi square value of 4.32, significant at p=.36.

RESPONSE	NO CHILDREN IN SCHOOL	NATIONAL TOTALS
Very Important	78	80
Fairly Important	13	15
Not VeryImportant	6	3
Not Important at All	2	1
Don't Know	1	1

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: The null hypothesis states that the proportion of responses those who live in cities and do not have school-age children and the national responses are same.

Alternative hypothesis: At least one of the null hypothesis statements is false.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square test for homogeneity.

Analyze sample data. Applying the chi-square test for homogeneity to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = (r - 1) * (c - 1) = (5 - 1) * (2 - 1)
D.F = 4
E_r,c = (n_r * n_c) / n

?² = 4.32

where DF is the degrees of freedom, r is the number of populations, c is the number of levels of the categorical variable, n_r is the number of observations from population r, n_c is the number of observations from level c of the categorical variable, n is the number of observations in the sample, E_r,c is the expected frequency count in population r for level c, and O_r,c is the observed frequency count in population r for level c.

We use the Chi-Square Distribution Calculator to find P(?² > 4.32) = 0.36.

Interpret results. Since the P-value (0.36) is greater than the significance level (0.05), we accept the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that there is a difference in responses between those who live in cities and do not have school-age children and the national responses.