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)	60	47	107
M.D.	67	95	162
Lawyer	51	86	137
Column Total	178	228	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?


State the null and alternate hypotheses.

H₀: Myers-Briggs preference and profession are independent
H₁: Myers-Briggs preference and profession are not independent.

H₀: Myers-Briggs preference and profession are not independent
H₁: Myers-Briggs preference and profession are not independent.   

H₀: Myers-Briggs preference and profession are not independent
H₁: Myers-Briggs preference and profession are independent.

H₀: Myers-Briggs preference and profession are independent
H₁: Myers-Briggs preference and profession are independent.

(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.)


Are all the expected frequencies greater than 5?

What sampling distribution will you use?

uniform

Student's t

normal

binomial

chi-square

What are the degrees of freedom?


(c) Find or estimate the P-value of the sample test statistic.

p-value > 0.1000

.050 < p-value < 0.100  

  0.025 < p-value < 0.0500

.010 < p-value < 0.0250

.005 < p-value < 0.010

p-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence?

Since the P-value > α, we fail to reject the null hypothesis.

Since the P-value > α, we reject the null hypothesis.    

Since the P-value ≤ α, we reject the null hypothesis.

Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, there is insufficient evidence to conclude that Myers-Briggs preference and the profession are not independent.

At the 5% level of significance, there is sufficient evidence to conclude that Myers-Briggs preference and the profession are not independent.    

Answer 1

(a) Level of significance: 0.05

State the null and alternate hypotheses.

H₀: Myers-Briggs preference and profession are independent
H₁: Myers-Briggs preference and profession are not independent.

(b)

Observed Frequencies
	E	I	Total
Clergy	60	47	107
M.D.	67	95	162
Lawyer	51	86	137
Total	178	228	406
Expected Frequencies
	E	I	Total
Clergy	178 * 107 / 406 = 46.9113	228 * 107 / 406 = 60.0887	107
M.D.	178 * 162 / 406 = 71.0246	228 * 162 / 406 = 90.9754	162
Lawyer	178 * 137 / 406 = 60.064	228 * 137 / 406 = 76.936	137
Total	178	228	406
	(fo-fe)²/fe
Clergy	(60 - 46.9113)²/46.9113 = 3.6519	(47 - 60.0887)²/60.0887 = 2.851
M.D.	(67 - 71.0246)²/71.0246 = 0.2281	(95 - 90.9754)²/90.9754 = 0.178
Lawyer	(51 - 60.064)²/60.064 = 1.3678	(86 - 76.936)²/76.936 = 1.0679

Test statistic:  

χ² = ∑ ((fo-fe)²/fe) = 9.345

All the expected frequencies are greater than 5.

Sampling distribution : chi-square

degrees of freedom, df = (r-1)(c-1) = 2

(c) p-value = CHISQ.DIST.RT(9.3446, 2) = 0.0094  

0.005 < p-value < 0.010

(d) Since the P-value ≤ α, we reject the null hypothesis.

(e) Conclusion :

At the 5% level of significance, there is sufficient evidence to conclude that Myers-Briggs preference and the profession are not independent.