Question

The city police chief wants to know the perceptions African-Americans have of the police force in his city. In comparison to white perception in the community, this information will tell the police chief if he has a community relations problem with the African American community that needs to be addressed. A survey reveals the following information. What would you tell the police chief given these results:

Opinion	African-American	White
Feel the police do a good job	74	223
Do not feel the police do a good job	76	7

Answer 1

We will check for the independence of the attributes if the class affects the opinion of people about the police

The following cross-tabulation have been provided. The row and column total have been calculated and they are shown below:

	African-American	White	Total
Feel the police do a good job	74	223	297
Do not feel the police do a good job	76	7	83
Total	150	230	380

The expected values are computed in terms of row and column totals. In fact, the formula is , where R_i​ corresponds to the total sum of elements in row i, C_j corresponds to the total sum of elements in column j, and T is the grand total. The table below shows the calculations to obtain the table with expected values:

Expected Values	African-American	White	Total
Feel the police do a good job	117.237	179.763	297
Do not feel the police do a good job	32.763	50.237	83
Total	150	230	380

Based on the observed and expected values, the squared distances can be computed according to the following formula:. The table with squared distances is shown below:

Squared Distances	African-American	White
Feel the police do a good job	15.946	10.399
Do not feel the police do a good job	57.059	37.212

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

H_0​: The two variables are independent

H_a​: The two variables are dependent

This corresponds to a Chi-Square test of independence.

Rejection Region

Based on the information provided, the significance level is α = 0.05 , the number of degrees of freedom is df = (2 - 1*(2 - 1) = 1.

Test Statistics

The Chi-Squared statistic is computed as follows:

= 120.616

Decision about the null hypothesis

Since it is observed that = 120.616 > = 3.841, it is then concluded that the null hypothesis is rejected.

Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the two variables are dependent, at the 0.05 significance level.

Hence, the opinion differs for the different groups.