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)	58	49	107
M.D.	70	92	162
Lawyer	52	85	137
Column Total	180	226	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?

Yes No   

What sampling distribution will you use?

normal

uniform

Student's t

binomial

chi-square

What are the degrees of freedom?


(c) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.)


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

a) Since the P-value > ?, we fail to reject the null hypothesis.

b) Since the P-value > ?, we reject the null hypothesis.    

c) Since the P-value ? ?, we reject the null hypothesis.

d) Since the P-value ? ?, we fail to reject the null hypothesis.

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

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

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

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

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

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square test for independence.

Analyze sample data. Applying the chi-square test for independence to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = (r - 1) * (c - 1) = (2 - 1) * (3 - 1)
D.F = 2
E_r,c = (n_r * n_c) / n


?² = 7

where DF is the degrees of freedom.

The P-value is the probability that a chi-square statistic having 2 degrees of freedom is more extreme than 7.0.

We use the Chi-Square Distribution Calculator to find P(?² > 7.0) = 0.0025

Interpret results. Since the P-value (0.0025) is less than the significance level (0.05), we cannot accept the null hypothesis.

Thus, we conclude that there is a relationship between Myers-Briggs preference and profession.

d) c) Since the P-value ? ?, we reject the null hypothesis.

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