Question

using chapter 13 data set 2, the researchers want to find out whether there is a difference among the graduation rates (and these are percentages) of five high schools over a 10-year period. Is there? (hint: are the years a factor?)

	High School 1	High School 2	High School 3	High School 4	High School 5
2003	67	82	94	65	88
2004	68	87	78	65	87
2005	65	83	81	45	86
2006	68	73	76	57	88
2007	67	77	75	68	89
2008	71	74	81	76	87
2009	78	76	79	77	81
2010	76	78	89	72	78
2011	72	76	76	69	89
2012	77	86	77	58	87

Answer 1

We will conduct one-way ANOVA test to find out whether there is a difference among the graduation rates of five high schools over a 10-year period.

For the given problem,

Experimental Unit - Students of High schools
Response variable - Graduation rates
Factor - High Schools (Year is not a factor here. It is replicate or samples for each high school)
Levels of factor - High School 1, High School 2, High School 3, High School 4, High School 5

Null hypotheses H0 : The average graduation rate is equal across all high schools.

Alternative hypotheses H1 : At least one average graduation rate is different for a high school.

We assume, Level of significance = 0.05

Degree of freedom of group = Number of level - 1 = 5 - 1 = 4

Degree of freedom of error = Number of observations - Number of level = 5 * 10 - 5 = 45

Critical value of F at DF = 4, 45 is 2.58

d.

Source	DF	SS	MS	F
Group	4	2733.28	683.32	17.728
Error	45	1734.5	38.54444
Total	49	4467.78

Let T_i be the total graduation rate for high school i, n_i be number of observations of high school i.

Let G be the total graduation rate of all observations and N be total number of observations.

X² is sum of squares for all the observations.

T1 = 709, T2 = 792 , T3 = 806, T4 = 652, T5 = 860

G = 709 + 792 + 806 + 652 + 860 = 3819

X² = 50465 + 62948 + 65310 + 43362 + 74078 = 296163

SST = X² - G²/N = 296163 - 3819²/50 = 4467.78

SSTR = T²/n - G²/N = (709² /10 + 792² /10 + 806² /10 + 652² /10 + 860² /10  ) - 3819²/50 = 2733.28

SSE = 4467.78 - 2733.28 = 1734.5

MSTR = SSTR / Df for group = 2733.28 / 4 = 683.32

MSE = SSE / Df for error = 1734.5 / 45 = 38.54444

F = MSTR / MSE = 683.32/ 38.54444 = 17.728

As, the observed value of F (17.728) is greater than the critical value (2.58), we reject the null hypothesis and conclude that the at least one average graduation rate is different for a high school. Thus, there is significant evidence that there is a difference among the graduation rates of five high schools over a 10-year period