To determine which opinion contributes the most to the chi-square test statistic, we need to perform a step-by-step analysis using the provided survey data.
### Step 1: Determine the Total Sample Size
First, we need to find the total count of survey responses:
[tex]\[
\text{Total Count} = \text{Believe Count} + \text{Do Not Believe Count} + \text{No Opinion Count}
\][/tex]
[tex]\[
397 + 254 + 367 = 1018
\][/tex]
### Step 2: Compute the Expected Count Under Equal Distribution
Next, we assume that if the opinions were equally distributed, each category would have the same number of responses. This expected count can be found by dividing the total count by the number of categories (which is 3):
[tex]\[
\text{Expected Count} = \frac{\text{Total Count}}{3}
\][/tex]
[tex]\[
\text{Expected Count} = \frac{1018}{3} = 339.33 \quad (\text{rounded to two decimal places})
\][/tex]
### Step 3: Calculate the Chi-Square Test Statistic Components
We calculate the chi-square test statistic for each opinion category using the formula:
[tex]\[
\chi^2 = \frac{(\text{Observed Count} - \text{Expected Count})^2}{\text{Expected Count}}
\][/tex]
Let’s compute the chi-square components for each opinion:
For "Believe":
[tex]\[
\chi^2_{\text{Believe}} = \frac{(397 - 339.33)^2}{339.33} \approx 9.8
\][/tex]
For "Do Not Believe":
[tex]\[
\chi^2_{\text{Do Not Believe}} = \frac{(254 - 339.33)^2}{339.33} \approx 21.46
\][/tex]
For "No Opinion":
[tex]\[
\chi^2_{\text{No Opinion}} = \frac{(367 - 339.33)^2}{339.33} \approx 2.26
\][/tex]
### Step 4: Identify the Opinion Contributing the Most
To find out which opinion contributes the most to the chi-square test statistic, we compare the chi-square components calculated:
[tex]\[
\chi^2_{\text{Believe}} \approx 9.8
\][/tex]
[tex]\[
\chi^2_{\text{Do Not Believe}} \approx 21.46
\][/tex]
[tex]\[
\chi^2_{\text{No Opinion}} \approx 2.26
\][/tex]
The largest chi-square value is for "Do Not Believe," which is approximately 21.46. Therefore, this is the opinion that contributes the most to the chi-square test statistic.
### Step 5: Determine if Observed Count is Larger or Smaller Than Expected
For the "Do Not Believe" category, compare the observed count to the expected count:
[tex]\[
\text{Observed Count of "Do Not Believe"} = 254
\][/tex]
[tex]\[
\text{Expected Count} = 339.33
\][/tex]
Since the observed count (254) is less than the expected count (339.33), the observed count for "Do Not Believe" is smaller than we would expect.
### Conclusion
- The opinion that contributes the most to the chi-square test statistic is "Do Not Believe."
- For this opinion, the observed count is smaller than the expected count.
