To calculate the conditional relative frequencies for students who prefer sprinkles or jelly beans, given that they prefer chocolate or vanilla, we will use the data from the original two-way frequency table.
The total number of students who prefer each ice cream flavor (Vanilla and Chocolate) will be used as the denominators for the conditional relative frequency calculations.
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Vanilla:
- Total votes for Vanilla = 175
- Votes for Sprinkles (Vanilla) = 102
- Votes for Jelly Beans (Vanilla) = 73
The conditional relative frequency for both toppings when given Vanilla is calculated as follows:
- Sprinkles (Vanilla): \[ \text{Relative Frequency (Sprinkles | Vanilla)} = \left( \frac{102}{175} \right) \times 100 \approx 58.29% \quad \text{(rounded to 58%)} \]
- Jelly Beans (Vanilla): \[ \text{Relative Frequency (Jelly Beans | Vanilla)} = \left( \frac{73}{175} \right) \times 100 \approx 41.71% \quad \text{(rounded to 42%)} \]
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Chocolate:
- Total votes for Chocolate = 190
- Votes for Sprinkles (Chocolate) = 65
- Votes for Jelly Beans (Chocolate) = 125
The conditional relative frequency for both toppings when given Chocolate is calculated as follows:
- Sprinkles (Chocolate): \[ \text{Relative Frequency (Sprinkles | Chocolate)} = \left( \frac{65}{190} \right) \times 100 \approx 34.21% \quad \text{(rounded to 34%)} \]
- Jelly Beans (Chocolate): \[ \text{Relative Frequency (Jelly Beans | Chocolate)} = \left( \frac{125}{190} \right) \times 100 \approx 65.79% \quad \text{(rounded to 66%)} \]
Now we can summarize the results in a two-way table of conditional relative frequencies:
| | Sprinkles (%) | Jelly Beans (%) | |-------------|---------------|------------------| | Vanilla | 58% | 42% | | Chocolate | 34% | 66% |
So, the completed conditional relative frequency table would be:
Sprinkles (%): 58
Jelly Beans (%): 42