Q.12 - The Practice of Statistics for AP

Question

Housing According to the Census Bureau, the distribution by ethnic background of the New York City population in a recent year was 

The manager of a large housing complex in the city wonders whether the distribution by the race of the complex’s residents is consistent with the population distribution. To find out, she records data from a random sample of $800$ residents. The table below displays the sample data

Are these data significantly different from the city’s distribution by race? Carry out an appropriate test at the $\alpha =0.05$ level to support your answer. If you find a significant result, perform follow-up analysis.

Step-by-Step Solution

Answer

From the given information, 

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

Step 1: Given Information

It is given in the question that, the distribution by ethnic background of the New York City population in a recent year was  

The table below displays the sample data

Step 2: Explanation

The null and alternative hypotheses are: 

$H_0:p_1=0.28,p_2=0.24,p_3=0.35,p_4=0.12,p_5=0.01$

$H_a:\text{At least one of}{p}_i\text{is different}$

The calculation for test statistics is done as:

The test statistic is :

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

     $=26.0625$

 The degree of freedom is calculated as: 

$\text{Degree of freedom}=\text{Number of categories}-1$

                                   $=5-1$

                                    $=4$

The p-value using the chi-square table at $4$ degrees of freedom is $0.00003$

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