Q2 A principal at a small school wanted to know if the racia

Q2) A principal at a small school wanted to know if the racial makeup of the school staff mirrored that of the student body. The student body broke down into the following racial composition: 31 were White, 18 were Latino, 10 were African American, and 6 were other races. Of the staff members, 42 were White, 4 were Latino, 15 were African American, and 4 were other races. SHOW ALL WORK STARTING WITH FORMULA

a) Conduct a Chi-Square Test for Goodness of Fit to determine if the racial makeup of the staff members is different from that of the students. This can be done by treating the racial composition of the students as the expected frequency for the observed frequency of the racial composition of the staff. (Use a .05 significance level and make sure you show all 5 steps of the hypothesis test)

Scoring Rubric

State the hypothesis Deduct as follows:

If any aspect of null hypothesis written in words is incorrect (.5 point)

If any aspect of null hypothesis written in notation is incorrect (.5 point)

If any aspect of alternative hypothesis written in words is incorrect (.5 point)

If any aspect of alternative hypothesis written in notation is incorrect (.5 point)

Determine sampling distribution

It is critical that students choose a Chi Square (0 points)

Determine critical value Deduct as follows:

If the critical value is not correct (1 point)

If degrees of freedom are not correct (1 point)

Conduct hypothesis test Deduct as follows:

If correct equation is not used (2 points)

If correct X² statistic is not found (2 points)

If X² obtained is not compared to X² critical in some form – words or notation (1 point)

Make decision about rejecting/failing to reject the hypothesis Deduct as follows:

If student does not explicitly state “reject the null hypothesis” (1 point)

If student does not write correct conclusion in words (1 point)

If student does not write correct conclusion in notation (1 point)

Solution

Ho: The racial distrbution of the school staff is the same as the student body.

Ha: The racial distrbution of the school staff is not the same as the student body.

***********************************

We use a chi^2 distrbution here.

Doing an Expected Value Chart,          

Body   Staff      
36.5   36.5  
11   11  
12.5   12.5  
5   5  

With df = (a - 1)(b - 1), where a and b are the number of categories of each variable,          
          
a =    2      
b =    4      
          
df =    3      
          
Thus, the critical value is          
          
significance level =    0.05      
          
chi^2(critical) =    7.814727903   [ANSWER, CRITICAL VALUE]  

          
Using chi^2 = Sum[(O - E)^2/E],          
          
chi^2 =    11.96662516   [ANSWER, CHI^2 STATISTIC]  
          

As out chi^2 > 7.8147, then WE REJECT THE NULL HYPOTHESIS.
          
Thus, there is significant evidence that the racial distrbution of the school staff is not the same as the student body.

***********************************

Please write the notation form of the hypotheses and conclusion as you do in class, as not everybody has the same notation.