### Analysis of Baseball Fans' Opinions on the New Commissioner's Performance In a study involving 100 baseball fans from four regions of the country, opinions were collected on the new commissioner's job performance. The table below summarizes the observed and expected values, along with the Chi-squared (\( \chi^2 \)) contributions from each region. #### Hypotheses - **\( H_0 \):** The variables are independent. - **\( H_a \):** At least two variables are dependent (claim). #### Observed Values | Region | Favorable | Unfavorable | No Opinion | |------------|-----------|-------------|------------| | Northeast | 41 | 29 | 34 | | Southeast | 44 | 28 | 32 | | Midwest | 57 | 26 | 21 | | West | 49 | 25 | 30 | #### Expected Values | Region | Favorable | Unfavorable | No Opinion | |------------|-----------|-------------|------------| | Northeast | 47.75 | 27 | 29.25 | | Southeast | 47.75 | 27 | 29.25 | | Midwest | 47.75 | 27 | 29.25 | | West | 47.75 | 27 | 29.25 | #### \( R^2 \) Contributions | Region | Favorable | Unfavorable | No Opinion | |------------|-----------|-------------|------------| | Northeast | 0.954 | 0.148 | 0.771 | | Southeast | 0.295 | 0.037 | 0.259 | | Midwest | 1.792 | 0.037 | 2.327 | | West | 0.033 | 0.148 | 0.019 | ### Statistical Decision The question posed is whether there is enough evidence to reject \( H_0 \) and support the claim of dependency at the 0.10 level of significance. #### Options for Decision A. Yes, the test statistic \( \chi^2 = 10.645 \) is in the rejection region with critical value 6.82. B.
### Analysis of Baseball Fans' Opinions on the New Commissioner's Performance In a study involving 100 baseball fans from four regions of the country, opinions were collected on the new commissioner's job performance. The table below summarizes the observed and expected values, along with the Chi-squared (\( \chi^2 \)) contributions from each region. #### Hypotheses - **\( H_0 \):** The variables are independent. - **\( H_a \):** At least two variables are dependent (claim). #### Observed Values | Region | Favorable | Unfavorable | No Opinion | |------------|-----------|-------------|------------| | Northeast | 41 | 29 | 34 | | Southeast | 44 | 28 | 32 | | Midwest | 57 | 26 | 21 | | West | 49 | 25 | 30 | #### Expected Values | Region | Favorable | Unfavorable | No Opinion | |------------|-----------|-------------|------------| | Northeast | 47.75 | 27 | 29.25 | | Southeast | 47.75 | 27 | 29.25 | | Midwest | 47.75 | 27 | 29.25 | | West | 47.75 | 27 | 29.25 | #### \( R^2 \) Contributions | Region | Favorable | Unfavorable | No Opinion | |------------|-----------|-------------|------------| | Northeast | 0.954 | 0.148 | 0.771 | | Southeast | 0.295 | 0.037 | 0.259 | | Midwest | 1.792 | 0.037 | 2.327 | | West | 0.033 | 0.148 | 0.019 | ### Statistical Decision The question posed is whether there is enough evidence to reject \( H_0 \) and support the claim of dependency at the 0.10 level of significance. #### Options for Decision A. Yes, the test statistic \( \chi^2 = 10.645 \) is in the rejection region with critical value 6.82. B.
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Transcribed Image Text:### Analysis of Baseball Fans' Opinions on the New Commissioner's Performance
In a study involving 100 baseball fans from four regions of the country, opinions were collected on the new commissioner's job performance. The table below summarizes the observed and expected values, along with the Chi-squared (\( \chi^2 \)) contributions from each region.
#### Hypotheses
- **\( H_0 \):** The variables are independent.
- **\( H_a \):** At least two variables are dependent (claim).
#### Observed Values
| Region | Favorable | Unfavorable | No Opinion |
|------------|-----------|-------------|------------|
| Northeast | 41 | 29 | 34 |
| Southeast | 44 | 28 | 32 |
| Midwest | 57 | 26 | 21 |
| West | 49 | 25 | 30 |
#### Expected Values
| Region | Favorable | Unfavorable | No Opinion |
|------------|-----------|-------------|------------|
| Northeast | 47.75 | 27 | 29.25 |
| Southeast | 47.75 | 27 | 29.25 |
| Midwest | 47.75 | 27 | 29.25 |
| West | 47.75 | 27 | 29.25 |
#### \( R^2 \) Contributions
| Region | Favorable | Unfavorable | No Opinion |
|------------|-----------|-------------|------------|
| Northeast | 0.954 | 0.148 | 0.771 |
| Southeast | 0.295 | 0.037 | 0.259 |
| Midwest | 1.792 | 0.037 | 2.327 |
| West | 0.033 | 0.148 | 0.019 |
### Statistical Decision
The question posed is whether there is enough evidence to reject \( H_0 \) and support the claim of dependency at the 0.10 level of significance.
#### Options for Decision
A. Yes, the test statistic \( \chi^2 = 10.645 \) is in the rejection region with critical value 6.82.
B.
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