Question

(Printed work preferable. Explain step-by-step please.)

1. An educational researcher interested in the consistency of school absenteeism over time studied a sample of 8 high school students for whom complete high school records were available. The researcher counted the number of days of school each student missed while in the sixth grade and how many they missed while in the tenth grade. He obtained the following results:

Student	# Days Missed in 6th Grade	# Days Missed in 10th
A	4	10
B	2	4
C	21	11
D	1	3
E	3	1
F	5	5
G	4	9
H	8	5

            

a. What is the strength and the direction of the association between the number of days missed in sixth grade and the number of days missed in tenth grade. Be sure to provide the statistic on which your answer is based.

b. Can the correlation be generalized to the larger population of students? (You will need to perform a hypothesis test.) Explain your answer and show all necessary work. Explain your answer. You should write out the research and null hypotheses in Greek notation.

Answer 1

The sample Coefficient of Correlation is a good statistics to measure the linear relationship between 2 variables.

Let X=# Days Missed in 6th Grade Y= # Days Missed in 10th

We get the following from the sample

n=8 is the sample size

the sample means are

The sum of squares are

The sample coefficient of correlation is

We will classify the strength of association as medium if the correlation coefficient is between 0.3 to 0.7. The positive sign indicates a positive direction of association.

ans: The strength of linear association between the number of days missed in sixth grade and the number of days missed in tenth grade, indicated by the correlation coefficient of 0.6087, is medium and the  direction of association is positive.

b) Let be the true correlation of the population of students. We want to test if the association between the number of days missed in sixth grade and the number of days missed in tenth grade can be generalized to this population.

That is we want to test the following hypotheses

The test statistics is

The degrees of freedom for t stat is df=n-2=8-2=6

This is a 2 tailed test (the alternative hypothesis has "not equal to")

The right tail critical value for is obtained using

Using the t table for df=6 and area under the right tail=0.025 (or the combined area under both tails=0.05), we get the right tail critical value = 2.447.

The critical values are -2.447 and +2.447

We will reject the null hypothesis if the test statistics doe not lie in the interval -2.447 to +2.447.

Here, the test statistics is 1.879 and it lies in the interval -2.447 to +2.447. Hence we do not reject the null hypothesis.

We can conclude that there is no sufficient evidence to say that there is a linear association between the number of days missed in sixth grade and the number of days missed in tenth grade.

ans: the correlation cannot be generalized to the larger population of students.