Concept explainers
A random sample of 748 voters in a large city was asked how they voted in the presidential election of 2012. Calculate chi square and the column percentages for each table below and write a brief report describing the significance of the relationships as well as the patterns you observe.
a. Presidential preference and gender
Gender | |||
Preference | Male | Female | Totals |
Romney | 165 | 173 | 338 |
Obama Totals |
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b. Presidential preference and race/ethnicity
Preference | Race/Ethnicity | |||||
White | Black | Latino | Totals | |||
Romney | 289 | 5 | 44 | 338 | ||
Obama Totals |
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c. Presidential preference by education
Preference | Education | ||||
Less than HS | HS Graduate | College Graduate | Post-Graduate Degree | Totals | |
Romney | 30 | 180 | 118 | 10 | 338 |
Obama Totals |
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d. Presidential preference by religion
Preference | Religion | |||||
Protestant | Catholic | Jewish | None | Other | Totals | |
Romney | 165 | 110 | 10 | 28 | 25 | 338 |
Obama Totals |
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(a)
To find:
The chi square value for the given information and test for its significance.
Answer to Problem 10.15P
Solution:
There is no significant relationship between the gender of the voter and the presidential preference.
The Percentage table suggests that though being not significant, both the genders prefer Obama over Romney.
Explanation of Solution
Given:
The given statement is,
A random sample of 748 voters in a large city was asked how they voted in the presidential election of 2012.
The given table of information is,
Preference | Gender | ||
Male | Female | Totals | |
Romney | 165 | 173 | 338 |
Obama | 200 | 210 | 410 |
Totals | 365 | 383 | 748 |
Approach:
The confidence interval is an interval estimate from the statistics of the observed data that might contain the true value of the unknown population parameter.
The five step model for hypothesis testing is,
Step 1. Making assumptions and meeting test requirements.
Step 2. Stating the null hypothesis.
Step 3. Selecting the sampling distribution and establishing the critical region.
Step 4. Computing test statistics.
Step 5. Making a decision and interpreting the results of the test.
Formula used:
For a chi square, the expected frequency
Where N is the total of frequencies.
The chi square statistic is given by,
Where
And
The degrees of freedom for the bivariate table is given as,
Where r is the number of rows and c is the number of columns.
Calculation:
From the given information,
The observed frequency is given as,
Substitute 338 for row marginal, 365 for column marginal and 748 for N in equation
Substitute 338 for row marginal, 383 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 365 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 383 for column marginal and 748 for N in equation
Consider the following table,
165 | 164.93 | 0.07 | 0.0049 | 0.00 | |
173 | 173.07 | 0.0049 | 0.00 | ||
200 | 200.07 | 0.0049 | 0.00 | ||
210 | 209.93 | 0.07 | 0.0049 | 0.00 | |
Total | 748 | 748 | 0 |
The value
Substitute 165 for
Squaring the above obtained result,
Divide the above obtained result by
Proceed in a similar manner to obtain rest of the values of
The chi square value is given as,
Thus, the chi square value is 0.00.
Follow the steps for chi square hypothesis testing.
Step 1. Making assumptions and meeting test requirements.
Model:
Independent random sampling.
Level of measurement is nominal.
Step 2. Stating the null hypothesis.
The statement of the null hypothesis is that there is no significant relationship between the gender of the voter and the presidential preference.
Thus, the null and the alternative hypotheses are,
Step 3. Selecting the sampling distribution and establishing the critical region.
The sampling distribution is chi square.
The level of significance is
Number of rows is 2 and the number of columns of 2.
The degrees of freedom is given by,
And area of critical region is
Step 4. Computing test statistics.
The obtained chi square value is 0.00.
Step 5. Making a decision and interpreting the results of the test.
Since,
This implies that, there is no significant relationship between the gender of the voter and the presidential preference.
The following table gives the column wise percentages of gender preferences..
Preference | Gender | |
Male | Female | |
Romany | 45.21% | 45.17% |
Obama | 54.79% | 54.83% |
Totals | 100% | 100% |
The columns of the table are computed as,
Substitute the values from the given table of information,
Proceed in a similar manner to obtain rest of the values of the table.
The above table suggests that though being not significant, both the genders prefer Obama over Romney.
Conclusion:
There is no significant relationship between the gender of the voter and the presidential preference.
The Percentage table suggests that though being not significant, both the genders prefer Obama over Romney.
(b)
To find:
The chi square value for the given information and test for its significance.
Answer to Problem 10.15P
Solution:
There is a significant relationship between the race and ethnicity of the voter and the presidential preference.
The Percentage table suggests that mainly blacks prefer Obama with 95% majority.
Explanation of Solution
Given:
The given statement is,
A random sample of 748 voters in a large city was asked how they voted in the presidential election of 2012.
The given table of information is,
Preference | Race/Ethnicity | |||
White | Black | Latino | Totals | |
Romney | 289 | 5 | 44 | 338 |
Obama | 249 | 95 | 66 | 410 |
Totals | 538 | 100 | 110 | 748 |
Approach:
The confidence interval is an interval estimate from the statistics of the observed data that might contain the true value of the unknown population parameter.
The five step model for hypothesis testing is,
Step 1. Making assumptions and meeting test requirements.
Step 2. Stating the null hypothesis.
Step 3. Selecting the sampling distribution and establishing the critical region.
Step 4. Computing test statistics.
Step 5. Making a decision and interpreting the results of the test.
Formula used:
For a chi square, the expected frequency
Where N is the total of frequencies.
The chi square statistic is given by,
Where
And
The degrees of freedom for the bivariate table is given as,
Where r is the number of rows and c is the number of columns.
Calculation:
From the given information,
The observed frequency is given as,
Substitute 338 for row marginal, 538 for column marginal and 748 for N in equation
Substitute 338 for row marginal, 100 for column marginal and 748 for N in equation
Substitute 338 for row marginal, 110 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 538 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 100 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 110 for column marginal and 748 for N in equation
Consider the following table,
289 | 234.11 | 45.89 | 2105.89 | 8.66 | |
5 | 45.19 | 161.24 | 35.74 | ||
44 | 49.71 | 32.60 | 0.66 | ||
249 | 294.89 | 2105.89 | 7.14 | ||
95 | 54.81 | 40.19 | 1615.24 | 29.47 | |
66 | 60.29 | 5.71 | 32.60 | 0.54 | |
Total | 748 | 748 | 0 |
The value
Substitute 289 for
Squaring the above obtained result,
Divide the above obtained result by
Proceed in a similar manner to obtain rest of the values of
The chi square value is given as,
Thus, the chi square value is 82.21.
Follow the steps for chi square hypothesis testing.
Step 1. Making assumptions and meeting test requirements.
Model:
Independent random sampling.
Level of measurement is nominal.
Step 2. Stating the null hypothesis.
The statement of the null hypothesis is that there is no significant relationship between the race and ethnicity of the voter and the presidential preference.
Thus, the null and the alternative hypotheses are,
Step 3. Selecting the sampling distribution and establishing the critical region.
The sampling distribution is chi square.
The level of significance is
Number of rows is 2 and the number of columns of 3.
The degrees of freedom is given by,
And area of critical region is
Step 4. Computing test statistics.
The obtained chi square value is 82.21.
Step 5. Making a decision and interpreting the results of the test.
Since,
This implies that, there is a significant relationship between the race and ethnicity of the voter and the presidential preference.
The following table gives the column wise percentages of gender preferences..
Preference | Race/Ethnicity | ||
White | Black | Latino | |
Romany | 53.72% | 5% | 40% |
Obama | 46.28% | 95% | 60% |
Totals | 100% | 100% | 100% |
The columns of the table are computed as,
Substitute the values from the given table of information,
Proceed in a similar manner to obtain rest of the values of the table.
The above table suggests that mainly blacks prefer Obama with 95% majority.
Conclusion:
There is a significant relationship between the race and ethnicity of the voter and the presidential preference.
The Percentage table suggests that mainly blacks prefer Obama with 95% majority.
(c)
To find:
The chi square value for the given information and test for its significance.
Answer to Problem 10.15P
Solution:
There is a significant relationship between the education level of the voter and the presidential preference.
The Percentage table suggests that, except the high school graduates, rest all the others prefer Obama over Romney.
Explanation of Solution
Given:
The given statement is,
A random sample of 748 voters in a large city was asked how they voted in the presidential election of 2012.
The given table of information is,
Preference | Education | ||||
Less than HS | HS Graduate | College Graduate | Post-Graduate Degree | Totals | |
Romney | 30 | 180 | 118 | 10 | 338 |
Obama | 35 | 120 | 218 | 37 | 410 |
Totals | 65 | 300 | 336 | 47 | 748 |
Apprach:
The confidence interval is an interval estimate from the statistics of the observed data that might contain the true value of the unknown population parameter.
The five step model for hypothesis testing is,
Step 1. Making assumptions and meeting test requirements.
Step 2. Stating the null hypothesis.
Step 3. Selecting the sampling distribution and establishing the critical region.
Step 4. Computing test statistics.
Step 5. Making a decision and interpreting the results of the test.
Formula used:
For a chi square, the expected frequency
Where N is the total of frequencies.
The chi square statistic is given by,
Where
And
The degrees of freedom for the bivariate table is given as,
Where r is the number of rows and c is the number of columns.
Calculation:
From the given information,
The observed frequency is given as,
Substitute 338 for row marginal, 65 for column marginal and 748 for N in equation
Substitute 338 for row marginal, 300 for column marginal and 748 for N in equation
Substitute 338 for row marginal, 336 for column marginal and 748 for N in equation
Substitute 338 for row marginal, 47 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 65 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 300 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 336 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 47 for column marginal and 748 for N in equation
Consider the following table,
30 | 29.37 | 0.63 | 0.40 | 0.01 | |
180 | 135.56 | 44.44 | 1974.91 | 14.57 | |
118 | 151.83 | 1144.47 | 7.54 | ||
10 | 21.24 | 126.34 | 5.95 | ||
35 | 35.63 | 0.40 | 0.01 | ||
120 | 164.44 | 1974.91 | 12.00 | ||
218 | 184.17 | 33.83 | 1144.47 | 6.21 | |
37 | 25.76 | 11.24 | 126.34 | 4.90 | |
Total | 748 | 748 | 0 |
The value
Substitute 30 for
Squaring the above obtained result,
Divide the above obtained result by
Proceed in a similar manner to obtain rest of the values of
The chi square value is given as,
Thus, the chi square value is 51.19.
Follow the steps for chi square hypothesis testing.
Step 1. Making assumptions and meeting test requirements.
Model:
Independent random sampling.
Level of measurement is nominal.
Step 2. Stating the null hypothesis.
The statement of the null hypothesis is that there is no significant relationship between the education level of the voter and the presidential preference.
Thus, the null and the alternative hypotheses are,
Step 3. Selecting the sampling distribution and establishing the critical region.
The sampling distribution is chi square.
The level of significance is
Number of rows is 2 and the number of columns of 4.
The degrees of freedom is given by,
And area of critical region is
Step 4. Computing test statistics.
The obtained chi square value is 82.21.
Step 5. Making a decision and interpreting the results of the test.
Since,
This implies that, there is a significant relationship between the education level of the voter and the presidential preference.
The following table gives the column wise percentages of gender preferences..
Preference | Education | |||
Less than HS | HS Graduate | College Graduate | Post-Graduate Degree | |
Romany | 46.15% | 60% | 35.12% | 21.28% |
Obama | 53.85% | 40% | 64.88% | 78.72% |
Totals | 100% | 100% | 100% | 100% |
The columns of the table are computed as,
Substitute the values from the given table of information,
Proceed in a similar manner to obtain rest of the values of the table.
The above table suggests that, except the high school graduates, rest all the others prefer Obama over Romney.
Conclusion:
There is a significant relationship between the education level of the voter and the presidential preference.
The Percentage table suggests that, except the high school graduates, rest all the others prefer Obama over Romney.
(d)
To find:
The chi square value for the given information and test for its significance.
Answer to Problem 10.15P
Solution:
There is a significant relationship between the religion of the voter and the presidential preference.
The Percentage table suggests that, except the voters from the Catholic religion, rest all the other religion voters prefer Obama over Romney.
Explanation of Solution
Given:
The given statement is,
A random sample of 748 voters in a large city was asked how they voted in the presidential election of 2012.
The given table of information is,
Preference | Religion | |||||
Protestant | Catholic | Jewish | None | Other | Totals | |
Romney | 165 | 110 | 10 | 28 | 25 | 338 |
Obama | 245 | 55 | 20 | 60 | 30 | 410 |
Totals | 410 | 165 | 30 | 88 | 55 | 748 |
Approach:
The confidence interval is an interval estimate from the statistics of the observed data that might contain the true value of the unknown population parameter.
The five step model for hypothesis testing is,
Step 1. Making assumptions and meeting test requirements.
Step 2. Stating the null hypothesis.
Step 3. Selecting the sampling distribution and establishing the critical region.
Step 4. Computing test statistics.
Step 5. Making a decision and interpreting the results of the test.
Formula used:
For a chi square, the expected frequency
Where N is the total of frequencies.
The chi square statistic is given by,
Where
And
The degrees of freedom for the bivariate table is given as,
Where r is the number of rows and c is the number of columns.
Calculation:
From the given information,
The observed frequency is given as,
Substitute 338 for row marginal, 410 for column marginal and 748 for N in equation
Substitute 338 for row marginal, 165 for column marginal and 748 for N in equation
Substitute 338 for row marginal, 30 for column marginal and 748 for N in equation
Substitute 338 for row marginal, 88 for column marginal and 748 for N in equation
Substitute 338 for row marginal, 55 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 410 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 165 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 30 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 88 for column marginal and 748 for N in equation
Substitute 410 for row marginal, 55 for column marginal and 748 for N in equation
Consider the following table,
165 | 185.27 | 410.87 | 2.22 | ||
110 | 74.56 | 35.44 | 1255.99 | 16.85 | |
10 | 13.56 | 12.67 | 0.93 | ||
28 | 39.76 | 138.30 | 3.48 | ||
25 | 24.85 | 0.15 | 0.02 | 0.00 | |
245 | 224.73 | 20.27 | 410.87 | 1.83 | |
55 | 90.44 | 1255.99 | 13.89 | ||
20 | 16.44 | 3.56 | 12.67 | 0.77 | |
60 | 48.24 | 11.76 | 138.30 | 2.87 | |
30 | 30.15 | 0.02 | 0.00 | ||
Total | 748 | 748 | 0 |
The value
Substitute 165 for
Squaring the above obtained result,
Divide the above obtained result by
Proceed in a similar manner to obtain rest of the values of
The chi square value is given as,
Thus, the chi square value is 42.84.
Follow the steps for chi square hypothesis testing.
Step 1. Making assumptions and meeting test requirements.
Model:
Independent random sampling.
Level of measurement is nominal.
Step 2. Stating the null hypothesis.
The statement of the null hypothesis is that there is no significant relationship between the religion of the voter and the presidential preference.
Thus, the null and the alternative hypotheses are,
Step 3. Selecting the sampling distribution and establishing the critical region.
The sampling distribution is chi square.
The level of significance is
Number of rows is 2 and the number of columns of 4.
The degrees of freedom is given by,
And area of critical region is
Step 4. Computing test statistics.
The obtained chi square value is 42.84.
Step 5. Making a decision and interpreting the results of the test.
Since,
This implies that, there is a significant relationship between the religion of the voter and the presidential preference.
The following table gives the column wise percentages of gender preferences..
Preference | Religion | ||||
Protestant | Catholic | Jewish | None | Other | |
Romany | 40.24% | 66.67% | 33.33% | 31.82% | 45.45% |
Obama | 59.75% | 33.33% | 66.67% | 68.18% | 54.55% |
Totals | 100% | 100% | 100% | 100% | 100% |
The columns of the table are computed as,
Substitute the values from the given table of information,
Proceed in a similar manner to obtain rest of the values of the table.
The above table suggests that, except the voters from the Catholic religion, rest all the other religion voters prefer Obama over Romney.
Conclusion:
There is a significant relationship between the religion of the voter and the presidential preference.
The Percentage table suggests that, except the voters from the Catholic religion, rest all the other religion voters prefer Obama over Romney.
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Chapter 10 Solutions
Essentials Of Statistics
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