(a) Suppose a survey determines the political orientation of 60 men in a certain
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Question
(a) Suppose a survey determines the political orientation of 60 men in a certain community: Left Middle Right Men 12 18 30 60 Among these men, calculate the proportion belonging to each political category. Then show that a Chi-squared Test of the null hypothesis of equal proportions leads to its rejection at the = .05 significance level. Conclusion? (b) Suppose the survey also determines the political orientation of 540 women in the same communitv: Left Middle Right Women 108 162 270 540 Among these women, calculate the proportion belonging to each political category. How do these proportions compare with those in (a)? Show that a Chi-squared Test of the null hypothesis of equal proportions Ho: 1eft 1 women-Zyid I women-Right I women leads to its rejection at the = .05 significance level. Conclusion?Explanation / Answer
Solutions:
Part a
First of all we have to find proportions of men belonging to each political category. Required proportions are given as below:
Left
Middle
Right
Total
Men
12
18
30
60
Proportion
12/60 = 0.2
18/60 = 0.3
30/60 = 0.5
1
Now, we have o use chi square test for testing the given null hypothesis of equal proportions.
Test statistic formula is given as below:
Chi square = [(O – E)^2/E]
Where O is observed frequencies and E is expected frequencies.
Table for calculation of test statistic is given as below:
O
E
(O - E)^2
(O - E)^2/E
Left
12
20
64
3.2
Middle
18
20
4
0.2
Right
30
20
100
5
Total
60
60
8.4
Chi square = [(O – E)^2/E] = 8.4
Degrees of freedom = n - 1 = 3 – 1 = 2
P-value = 0.014995577 (by using Chi square table)
We are given
= 0.05
P-value < = 0.05
So, we reject the null hypothesis of equal proportions.
The proportions of men for left, middle and right political categories are not same.
There is insufficient evidence to conclude that proportions of men for left, middle and right political categories are same.
Part b
Now, we have to perform same test for women. The required proportions and chi square test is given as below:
Left
Middle
Right
Total
Women
108
162
270
540
Proportion
0.2
0.3
0.5
1
O
E
(O - E)^2
(O - E)^2/E
Left
108
180
5184
28.8
Middle
162
180
324
1.8
Right
270
180
8100
45
Total
540
540
75.6
Chi square = [(O – E)^2/E] = 75.6
Degrees of freedom = n - 1 = 3 – 1 = 2
P-value = 0.00 (by using Chi square table)
We are given
= 0.05
P-value < = 0.05
So, we reject the null hypothesis of equal proportions.
The proportions of women for left, middle and right political categories are not same.
There is insufficient evidence to conclude that proportions of women for left, middle and right political categories are same.
Now, we have to check whether the two categorical variables such as gender and political category are independent from each other or not.
We assume 5% level of significance. = 0.05
The test statistic formula for this test is given as below:
Chi square = [(O – E)^2/E]
Where, O is observed frequencies and E is expected frequencies.
Expected frequencies are calculated as below:
E = Row total * Column total / Grand total
For checking the given hypothesis, table for calculations are given as below:
Observed Frequencies (O)
Political category
Gender
Left
Middle
Right
Total
Men
12
18
30
60
Women
108
162
270
540
Total
120
180
300
600
Expected Frequencies (E)
Political category
Gender
Left
Middle
Right
Total
Men
12
18
30
60
Women
108
162
270
540
Total
120
180
300
600
O - E
0
0
0
0
0
0
(O - E)^2/E
0
0
0
0
0
0
Chi square = [(O – E)^2/E] = 0.00
Number of rows = r = 2
Number of columns = c = 3
DF = (r – 1)*(c – 1) = (2 – 1) * (3 – 1) = 1*2 = 2
P-value = 1
P-value > = 0.05
So, we do not reject the null hypothesis.
P-value is 1, which means 100% acceptance at 5% level of significance.
We concluded that there is 100% acceptance that the proportions of men and women are independent from the political categories.
Left
Middle
Right
Total
Men
12
18
30
60
Proportion
12/60 = 0.2
18/60 = 0.3
30/60 = 0.5
1
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