Question

 

You want to know if owning cats is associated with having a job. Consider the following bivariate table that examines the relationship between owning cats and having a job.

	Own a Cat	Do not Own a Cat	Total
Have a Job	650	500	1150
Do not have a Job	200	150	350
Total	850	650	1500

a. Which measure of association is the most appropriate? Why?

b. Calculate the measure of association (the one you indicated in part a)

c.Interpret this measure of association for someone who is not familiar with statistics .

Answer 1

Which measure of association is the most appropriate? Why?

The most appropriate measure of the association will be the Chi-square statistic of the association because it will test how likely it is that an observed distribution is due to chance. It is also called a "goodness of fit" statistic, because it measures how well the observed distribution of data fits with the distribution that is expected if the variables are independent.

Calculate the measure of association (the one you indicated in part a)

	Own a Cat	Do not Own a Cat
Have a job	650	500
Do not have a job	200	150

For observed value 650, let us calculate the expected value:

Expected value = Row Total*Column Total/Total marginal count

= 1150*850/1500 = 651.67

For observed value 500, let us calculate the expected value:

Expected value = Row Total*Column Total/Total marginal count

= 1150*650/1500 = 498.33

For observed value 200, let us calculate the expected value:

Expected value = Row Total*Column Total/Total marginal count

= 850*350/1500 = 198.33

For observed value 150, let us calculate the expected value:

Expected value = Row Total*Column Total/Total marginal count

= 350*650/1500 = 151.67

Let us calculate the Observed value-Expected value(O-E) for observed value 650:

O-E = 650-651.67 = -1.67

Let us calculate the Observed value-Expected value(O-E) for observed value 500:

O-E = 500-498.33 = 1.67

Let us calculate the Observed value-Expected value(O-E) for observed value 200:

O-E = 200-198.33 = 1.67

Let us calculate the Observed value-Expected value(O-E) for observed value 150:

O-E = 150-151.67 = -1.67

Let us calculate the (Observed value-Expected value)²/Expected value{(O-E)²/E} for observed value 650:

(O-E)²/E = (-1.67)²/651.67 = 0.00

Let us calculate the (Observed value-Expected value)²/Expected value{(O-E)²/E} for observed value 500:

(O-E)²/E = (1.67)²/498.33 = 0.01

Let us calculate the (Observed value-Expected value)²/Expected value{(O-E)²/E} for observed value 200:

(O-E)²/E = (1.67)²/198.33 = 0.01

Let us calculate the (Observed value-Expected value)²/Expected value{(O-E)²/E} for observed value 150:

(O-E)²/E = (-1.67)²/151.67 = 0.02

The Chi-square statistic is the sum of all (Observed value-Expected value)²/Expected value.

Chi-square statistic = 0.00+0.01+0.01+0.02 = 0.04

The completed table is:

		Own a Cat	Do not Own a Cat	Total
Have a job	Observed	650	500	1150
	Expected	651.67	498.33	1150.00
	O - E	-1.67	1.67	0.00
	(O - E)² / E	0.00	0.01	0.01
Do not have a job	Observed	200	150	350
	Expected	198.33	151.67	350.00
	O - E	1.67	-1.67	0.00
	(O - E)² / E	0.01	0.02	0.03
Total	Observed	850	650	1500
	Expected	850.00	650.00	1500.00
	O - E	0.00	0.00	0.00
	(O - E)² / E	0.02	0.02	0.04
		.04	chi-square
		1	df
		.8373	p-value

Interpret this measure of association for someone who is not familiar with statistics.

The Chi-square statistic is 0.04

Also, the tabulated value of Chi-square distribution at 5% level of significance with 1 degree of freedom is 3.841

Since Chi-square calculated(0.04) is less than 3.841, thus we conclude that owning a cat is not associated with having a job.