Background

Sometime around 1965 a survey was conducted where students at a certain university were asked the following question. “Do you think that some racial and religious groups should be prevented from living in certain sections of cities?” A summary of their response are recorded in the following table. The region of the United States that the student respondent was from was also recorded.

Region Agree Undecided Disagree
East 89 79 297
Midwest 118 130 350
South 241 140 248
West 37 59 197

This analysis will be using a \(\chi^2\) comparison test to determine if an individual’s response is independent of the region a person is from. The two hypotheses for this analysis will be as follows:

\(H_0:\ \text{Region and response are not associated with each other}\)
\(H_a:\ \text{Region and response are associated with each other}\)

This analysis will be conducted with an \(\alpha\) of 0.05.

Analysis

Visualzation
palette(c("skyblue4","firebrick", "skyblue", "sienna1"))
barplot(discrim, beside = TRUE, legend.text = TRUE, xlab = "Response in Favor of Segregation", ylab = "Frequency of Response", main = "1965 College Segregation Survey", col = palette(), args.legend = list(cex = 0.89, bty = "n", x = "topleft", title = "Region"))

If the region and response are not associated with each other, we would expect the three distributions to have the same pattern. If you observe, the southern region of the US varies more in relative response compared to the other regions. At the very least, I suspect that the people from the South will contribute to rejecting our null hypothesis. Let’s see what our numeric data says on this observation.

Numerical Data

The numerical results of the \(\chi^2\) analysis are below:

Test statistic Df p value
125 6 1.476e-24

Since our the p-value of our \(\chi^2\) is less than our \(\alpha\) (0.05) we will reject our null hypothesis. In other words, we believe that there is enough evidence to suggest that the region a person is from will affect their response.

Interpretation: Was There Significant Discrimination in the South?

Listed below are the residuals of our \(\chi^2\) test. The more a Pearson residual strays from zero the more it contributes the a larger \(\chi^2\) value. As described below, people who responded from the South had residuals that were much different than zero. There were more people, from the South, who agreed with segregation than what was expected. There were also less people that disagreed with segregation than what was expected. I would suggest a further analysis into southern culture to determine what factors contributed to their region influencing their decision.

Region Agree Undecided Disagree
East -2.309 -1.696 2.575
Midwest -2.326 0.6392 1.159
South 7.043 0.9423 -5.27
West -4.088 -0.1577 2.821

I need to point out that the results of this analysis need to be used with a word of caution. Everyone who responded to the survey were all going to the same college and were likely to be in their early 20’s. The results of this study may not be an accurate representation of the country and should be used cautiously.