Question

The table below summarizes data from a survey of a sample of women. Using a 0.05 significance​ level, and assuming that the sample sizes of 700 men and 300 women are​ predetermined, test the claim that the proportions of​ agree/disagree responses are the same for subjects interviewed by men and the subjects interviewed by women. Does it appear that the gender of the interview affected the responses of​ women?

	Man	Woman
Woman who agree	481	245
Woman who disagree	219	55

A. Identify the null and alternative hypotheses.

B. Compute the test statistic.

C. Find the critical​ value(s).

D. What is the conclusion based on the hypothesis​ test?

Answer 1

Step-1: State the hypothesis:

H0: The proportions of​ agree/disagree responses are the same for subjects interviewed by men and the subjects interviewed by women.

H1: The proportions of​ agree/disagree responses are not the same for subjects interviewed by men and the subjects interviewed by women.


Step-2: Observed Frequencies:

	Man	Woman	Total
Agree	481	245	726
Disagree	219	55	274
Total	700	300	1000

Step-3: Expected Frequencies:

Fe11 = (726 x 700) / 1000 = 508.2

Fe12 = (726 x 300) / 1000 = 217.8

Fe21 = ( 274 x 700 ) / 1000 = 191.8

Fe22 = ( 274 x 300 ) / 1000 = 82.2

	Man	Woman	Total
Agree	508.2	217.8	726
Disagree	191.8	82.2	274
Total	700	300	1000

Step-4: Compute Chi-square:

χ2 = ∑ [ (Oi - Ei)² / Ei ]

= (481 - 508.2)² / 508.2 + (245 - 217.8)² / 217.8 + (219 - 191.8)² /191.8 + (55 -82.2)² / 82.2

= 1.4558 + 3.3969 + 3.8574 + 9.0005

= 17.7105

χ2 = 17.7105     -------------------------(Calculated value)

Step-5: Compute the degrees of freedom (df):
df= (2 - 1)⋅(2 - 1)

df = 1
Level of significance = 0.05

χ2 = 3.841     -------------------------(Tabulated value)

Step-6: Decision:

Calculated value > Tabulated value

17.7105   > 3.841

P value = 1 – (0.05/2)

             = 1 – 0.025

P value = 0.975

Critical value = 1.96

Step-7: Conclusion:

Null hypothesis is rejected.

Alternative hypothesis is accepted.

H1: The proportions of​ agree/disagree responses are not the same for subjects interviewed by men and the subjects interviewed by women.