To determine which opinion has the largest contribution to the chi-square test statistic and whether the observed count is smaller or larger than the expected count, follow these detailed steps:
1. Define the Observed Counts:
The survey respondents' counts for each category are:
- "Great deal" of confidence: 81
- "Fair amount" of confidence: 325
- "Not very much" confidence: 397
- "No confidence at all": 214
These counts sum up to 1,017, which is the total number of responses.
2. Calculate the Expected Counts:
If the four responses were equally likely, each response category would have an expected count. Given that there are 1,017 total responses and four categories, the expected count for each category is calculated as:
[tex]\[
\text{Expected Count} = \frac{\text{Total Responses}}{\text{Number of Categories}} = \frac{1017}{4} \approx 254.25
\][/tex]
Since the expected count needs to be an integer in practical situations, for simplicity, the expected count for each category can be considered:
[tex]\[
\text{Expected Count for each category} = 254
\][/tex]
So, the expected counts for each category are 254.
3. Calculate the Chi-Square Contributions:
The chi-square statistic for each category is computed using the formula:
[tex]\[
\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
\][/tex]
where [tex]\(O_i\)[/tex] are the observed counts and [tex]\(E_i\)[/tex] are the expected counts.
Let's break this down for each category:
- "Great deal" of confidence:
[tex]\[
\chi^2_1 = \frac{(81 - 254)^2}{254} \approx 117.83
\][/tex]
- "Fair amount" of confidence:
[tex]\[
\chi^2_2 = \frac{(325 - 254)^2}{254} \approx 19.85
\][/tex]
- "Not very much" confidence:
[tex]\[
\chi^2_3 = \frac{(397 - 254)^2}{254} \approx 80.51
\][/tex]
- "No confidence at all":
[tex]\[
\chi^2_4 = \frac{(214 - 254)^2}{254} \approx 6.30
\][/tex]
4. Determine the Largest Contribution:
Compare the chi-square contributions for each category:
[tex]\[
\chi^2_1 \approx 117.83, \quad \chi^2_2 \approx 19.85, \quad \chi^2_3 \approx 80.51, \quad \chi^2_4 \approx 6.30
\][/tex]
The largest contribution to the chi-square test statistic is about 117.83, which comes from the "Great deal" of confidence response category.
5. Compare Observed vs. Expected Counts:
For the "Great deal" of confidence category:
- Observed Count = 81
- Expected Count = 254
- Since 81 (Observed Count) is smaller than 254 (Expected Count), the observed count is smaller than the expected count.
In summary:
- The "Great deal" of confidence response has the largest contribution to the chi-square test statistic.
- For this age group, the observed count for the "Great deal" of confidence response is smaller than the expected count.
