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)	66	41	107
M.D.	64	98	162
Lawyer	61	76	137
Column Total	191	215	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 not independent
H₁: Myers-Briggs preference and profession are not independent.H₀: Myers-Briggs preference and profession are independent
H₁: Myers-Briggs preference and profession are 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 not 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?

YesNo    

What sampling distribution will you use?

normalchi-square    binomialStudent's tuniform

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?

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) The level of significance is given in the question as .

The correct null and alternative hypothesis are-

Myers-Briggs preference and profession are independent.

Myers-Briggs preference and profession are not independent.

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(b) The chi-square test-statistic is given as-

The formula to calculate the Expected frequency is :

Table for expected frequency:

Occupation	E	I
Clergy
M.D.
Lawyer

Now using the values for expected and observed frequency we calculate the chi-square statistic:

The chi-square statistic is calculated as

• YES: all the expected frequencies are greater than 5
• Sampling distribution: We use the Chi-square distribution for sampling.

Degrees of freedom:

Since, number of rows= 3 and number of columns=2

Hence,

________________________________________________________

(c) The p-value for the calculated sample test statistic, i.e. , with df= 2 is given by-

So, the p-value is 0.001

______________________________________________

(d) Decision: For the given significance level and for the , the decision is given as-

Since,

So, the p-value is less then the significance level, so our decision is to reject the null hypothesis.

__________________________________________

(e) Conclusion and Interpretation:

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

In other words, at the sample data provides enough evidence to support the alternative hypothesis, i.e., Myers-Briggs preference and profession are not independent.