Question

The following table shows the Myers-Briggs personality preferences for a random sample of 406 people in the listed professions. E refers to extroverted and I refers to introverted.

	Personality Type
Occupation	E	I	Row Total
Clergy (all denominations)	62	45	107
M.D.	69	93	162
Lawyer	57	80	137
Column Total	188	218	406

Use the chi-square test to determine if the listed occupations and personality preferences are independent at the 0.05 level of significance.

(a) What is the level of significance?

(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)

What are the degrees of freedom?

Answer 1

(a)

The level of significance is 0.05

(b)

Null hypothesis H0: The listed occupations and personality preferences are independent

Alternative hypothesis Ha: The listed occupations and personality preferences are not independent

The expected frequency counts are computed separately for each population at each level of the categorical variable, according to the following formula.

Er,c = (nr * nc) / n

where Er,c is the expected frequency count for population r at level c of the categorical variable, nr is the total number of observations from population r, nc is the total number of observations at treatment level c, and n is the total sample size.

Expected frequency table is,

Occupation	E	I	Row total
Clergy (all denominations)	(107 * 188)/406 = 49.547	(107 * 218)/406 = 57.453	107
M.D	(162 * 188)/406 = 75.015	(162 * 218)/406 = 86.985	162
Lawyer	(137 * 188)/406 = 63.438	(137 * 218)/406 = 73.562	137
Column Total	188	218	406

The test statistic is a chi-square random variable (Χ²) defined by the following equation.

Χ² = Σ [ (Or,c - Er,c)² / Er,c ]

where Or,c is the observed frequency count in population r for level c of the categorical variable, and Er,c is the expected frequency count in population r for level c of the categorical variable.

Χ² = (62 - 49.547)^2 / 49.547 + (45 - 57.453)^2 / 57.453 + (69 - 75.015)^2 / 75.015 + (93 - 86.985)^2 / 86.985 + (57 - 63.438)^2 / 63.438 + (80 - 73.562)^2 / 73.562

= 7.944

Degree of freedom (DF) is equal to:

DF = (r - 1) * (c - 1)

where r is the number of populations, and c is the number of levels for the categorical variable.

DF = (3 - 1) * (2 - 1) = 2

Critical value of Χ² at the 0.05 level of significance and df = 2 is 5.99

As, the observed chi-square statistic (7.944) is greater than the critical value, we reject the null hypothesis and conclude that there is significant evidence that the listed occupations and personality preferences are not independent at the 0.05 level of significance.