Question
The chef at a lunch café surveyed 150 teenagers and 300 adults on their side preferences when buying combo-meals. She groups the participants by age group and side preference. The results of the survey are in the two-way table below. Make a relative frequency table, then determine if there is evidence of an association between the two variables, age group and side preference.
Start by making a relative frequency table. What percentage of teenagers surveyed prefer salad?
Prefer
salad Prefer
soup Total
Teenagers 102 48 150
Adults 105 195 300
Total 207 243 450
Relative frequency table
Prefer
salad Prefer
soup Total
Teenagers
%
Adults
Start by making a relative frequency table. What percentage of teenagers surveyed prefer salad?
Prefer
salad Prefer
soup Total
Teenagers 102 48 150
Adults 105 195 300
Total 207 243 450
Relative frequency table
Prefer
salad Prefer
soup Total
Teenagers
%
Adults
Answers
GPT-4o mini
To create a relative frequency table, we will calculate the relative frequencies of each preference within both age groups (teenagers and adults) by dividing the counts by the total number of respondents in each age group.
The steps are as follows:
1. **Calculate the total for each age group.**
- Total teenagers = 150
- Total adults = 300
2. **Relative frequency for teenagers preferring salad:**
\[
\text{Relative frequency (Teenagers prefer salad)} = \frac{\text{Number of teenagers preferring salad}}{\text{Total teenagers}} = \frac{102}{150} \approx 0.68
\]
To find the percentage:
\[
0.68 \times 100 \approx 68\%
\]
3. **Relative frequency for teenagers preferring soup:**
\[
\text{Relative frequency (Teenagers prefer soup)} = \frac{\text{Number of teenagers preferring soup}}{\text{Total teenagers}} = \frac{48}{150} = 0.32
\]
4. **Relative frequency for adults preferring salad:**
\[
\text{Relative frequency (Adults prefer salad)} = \frac{\text{Number of adults preferring salad}}{\text{Total adults}} = \frac{105}{300} = 0.35
\]
5. **Relative frequency for adults preferring soup:**
\[
\text{Relative frequency (Adults prefer soup)} = \frac{\text{Number of adults preferring soup}}{\text{Total adults}} = \frac{195}{300} = 0.65
\]
Now, let's compile the relative frequency table:
| Preference | Teenagers (%) | Adults (%) |
|------------------|----------------|------------|
| Prefer salad | 68% | 35% |
| Prefer soup | 32% | 65% |
| Total | 100% | 100% |
### Conclusion
Percentage of teenagers surveyed who prefer salad is **68%**.
### Evidence of Association
To check for evidence of an association between age group and side preference, we can look at the distribution of preferences:
- Teenagers prefer salad significantly more (68%) compared to adults (35%).
- This disparity suggests there may be an association between age group and side preference because the preferences are not uniformly distributed between the age groups.
A Chi-square test could be conducted to statistically determine if this observed association is significant. However, even without performing the test, the difference in preferences indicates there is a likely association between age group and side preference.
The steps are as follows:
1. **Calculate the total for each age group.**
- Total teenagers = 150
- Total adults = 300
2. **Relative frequency for teenagers preferring salad:**
\[
\text{Relative frequency (Teenagers prefer salad)} = \frac{\text{Number of teenagers preferring salad}}{\text{Total teenagers}} = \frac{102}{150} \approx 0.68
\]
To find the percentage:
\[
0.68 \times 100 \approx 68\%
\]
3. **Relative frequency for teenagers preferring soup:**
\[
\text{Relative frequency (Teenagers prefer soup)} = \frac{\text{Number of teenagers preferring soup}}{\text{Total teenagers}} = \frac{48}{150} = 0.32
\]
4. **Relative frequency for adults preferring salad:**
\[
\text{Relative frequency (Adults prefer salad)} = \frac{\text{Number of adults preferring salad}}{\text{Total adults}} = \frac{105}{300} = 0.35
\]
5. **Relative frequency for adults preferring soup:**
\[
\text{Relative frequency (Adults prefer soup)} = \frac{\text{Number of adults preferring soup}}{\text{Total adults}} = \frac{195}{300} = 0.65
\]
Now, let's compile the relative frequency table:
| Preference | Teenagers (%) | Adults (%) |
|------------------|----------------|------------|
| Prefer salad | 68% | 35% |
| Prefer soup | 32% | 65% |
| Total | 100% | 100% |
### Conclusion
Percentage of teenagers surveyed who prefer salad is **68%**.
### Evidence of Association
To check for evidence of an association between age group and side preference, we can look at the distribution of preferences:
- Teenagers prefer salad significantly more (68%) compared to adults (35%).
- This disparity suggests there may be an association between age group and side preference because the preferences are not uniformly distributed between the age groups.
A Chi-square test could be conducted to statistically determine if this observed association is significant. However, even without performing the test, the difference in preferences indicates there is a likely association between age group and side preference.