Question #1 Since 1935, the Gallup organization has polled the U.S. public on a
ID: 3229911 • Letter: Q
Question
Question #1 Since 1935, the Gallup organization has polled the U.S. public on a number of political, economic, social, and lifestyle issues. One social issue that has received repeated attention is the public's perception of gender differences. In March 2017, Gallup polled United States residents on whether they perceived certain personality characteristics as more true of men or women. The table below provides the percentages of men and women who judged the characteristic easy-going as more true for one gender than the other. Easy-Going Men Women 60% More true of Men More true of Women 40% 48% Gallup randomly polled 360 men and 440 women in March 2017. The margin of error overall is t3.5 percentage points (t 5.0 for men and t 4.5 for Perform a goodness-of-fit test e., a one-sample chi square test) to determine whether men are more likely to believe that one gender is more easy going than the other. Note that for this and subsequent tests you must change the numbers in the table from percentages to frequenciesExplanation / Answer
Solution:
Here, we have to use the chi square test for independence of two categorical variables. First of all we have to change the percentages into frequencies which are given as below:
Men
Women
Total
More true of Men
360*0.60 = 216
440*0.52 = 229
Approximately
445
More true of Women
360*0.40 = 144
440*0.48 = 211 approximately
355
Total
360
440
800
The null and alternative hypothesis for this test is given as below:
Null hypothesis: H0: The two categorical variables easy going and gender are independent from each other.
Alternative hypothesis: Ha: The two categorical variables easy going and gender are not independent from each other.
We consider the level of significance or alpha value for this test as 5% or 0.05.
The test statistic formula for this test is given as below:
Chi square = [(O – E)^2/E]
Where, O = observed frequencies and E is expected frequencies.
Expected frequencies are calculated as below:
E = row total*column total / Grand total
Part g
We are given
Number of rows = r = 2
Number of columns = c = 2
Degrees of freedom = (r – 1)*(c – 1) = (2 – 1)*(2 – 1) = 1*1 = 1
= 0.05
Critical value = 3.841459
Decision rule: Reject H0 if Chi square test statistic > 3.841459, otherwise do not reject H0 at 5% level of significance.
The calculation tables for this test are given as below:
Observed Frequencies
Column variable
Row variable
Men
Women
Total
More true of Men
216
229
445
More true of Women
144
211
355
Total
360
440
800
Expected Frequencies
Column variable
Row variable
Men
Women
Total
More true of Men
200.25
244.75
445
More true of Women
159.75
195.25
355
Total
360
440
800
Part h
From above tables, we have
(O - E)
15.75
-15.75
-15.75
15.75
(O - E)^2/E
1.238764
1.013534
1.552817
1.270487
Chi square test statistic = [(O – E)^2/E]
Chi square test statistic = 1.238764 + 1.013534 + 1.552817 + 1.270487 = 5.075602
Part i
Critical value = 3.841459
Chi square test statistic > Critical value, so reject the null hypothesis H0
P-value = 0.024265
Alpha value = 0.05
P-value < Alpha value, so reject the null hypothesis H0
There is insufficient evidence to conclude that the two categorical variables easy going and gender are independent from each other.
Men
Women
Total
More true of Men
360*0.60 = 216
440*0.52 = 229
Approximately
445
More true of Women
360*0.40 = 144
440*0.48 = 211 approximately
355
Total
360
440
800