Question

A professor tests whether the loudness of noise during an exam (low, medium, and high) is independent of exam grades (pass, fail). The following table shows the observed frequencies for this test.

Noise Level

Low Medium High

Exam Pass 20 18 8 46

Fail 8 6 10 24

28 24 18 N=70

(a) Conduct a chi-square test for independence at a 0.05 level of significance. (Round your answer to two decimal places.)

x^2{obt}=

(b) Compute effect size using Cramer's V. (Round your answer to two decimal places.)

V =

Answer 1

In order to test , whether the loudness of noise during an exam (low, medium, and high) is independent of exam grades (pass, fail) we define the test hypothesis as:

Null hypothesis:

Test grade and noise level are independent.

Against the alternative hypothesis is

Test grade and noise level are not independent.

Observed data (frequency) are as follows:

		Noise Level
		Low	Medium	High
exam	Pass	20	18	8	46
	Fail	8	6	10	24
		28	24	18	70

Now the expected data (frequency) are calculated as:

Expected frequency=(Row total X Colunm total) / Grand total

		Noise Level
		Low	Medium	High	Sum
exam	Pass	18.4	15.77143	11.82857	46
	Fail	9.6	8.228571	6.171429	24
		28	24	18	70

To calculate the Chi square test statistics we define the chi square as:

		Noise Level
		Low	Medium	High	Sum
exam	Pass	21.73913	20.54348	5.410628	47.69324
	Fail	6.666667	4.375	16.2037	27.24537
	sum	28.4058	24.91848	21.61433	74.93861

=74.93861-70

=4.93861

Now as it is a contingency table view , therefore the degree of freedom of statistics is;

Therefore the critical value of test statistics at level of significance is:

from the standard chi square table .

in order to determined the P value for corresponding Chi square we follow Excell as:

=CHIDIST(4.9386,2)

Therefore ,

and corresponding

We fail to reject the null hypothesis and therefore we retain the null hypothesis.

b) Cramer’s V is measure the inter correlation between two contingency ( discrete variables). It is use to post test of strengths of association after chi-square significance for more than 2x2 rows and columns.

It is denoted as:

r: row and c: column

r: row and c: column

=0.07055

Therefore the effect size by cramer's V is :

V=0.07