Let's analyze the given data and the inferences from the survey step-by-step.
1. Calculate total preferences for each fruit in both samples:
- Total people who prefer apples: [tex]\(40 + 43 = 83\)[/tex]
- Total people who prefer oranges: [tex]\(64 + 63 = 127\)[/tex]
- Total people who prefer bananas: [tex]\(21 + 19 = 40\)[/tex]
2. Inference 1: About twice as many people prefer apples as prefer bananas.
- We compare the number of people who prefer apples to twice the number of people who prefer bananas: [tex]\(83\)[/tex] apples vs [tex]\(2 \times 40 = 80\)[/tex] bananas.
- Since [tex]\(83 \geq 80\)[/tex], this inference is True.
3. Inference 2: Bananas are preferred much less frequently than apples or oranges.
- We compare the preferences:
- Apples: 83
- Oranges: 127
- Bananas: 40
- Bananas are clearly less frequent than both apples and oranges. Thus, this inference is True.
4. Inference 3: Exactly twice as many people prefer oranges as prefer apples.
- We compare the number of people who prefer oranges to twice the number of people who prefer apples: [tex]\(127\)[/tex] oranges vs [tex]\(2 \times 83 = 166\)[/tex] apples.
- Since [tex]\(127 \neq 166\)[/tex], this inference is False.
5. Inference 4: Exactly three times as many people prefer bananas as prefer oranges.
- We compare the number of people who prefer bananas to three times the number of people who prefer oranges: [tex]\(40\)[/tex] bananas vs [tex]\(3 \times 127 = 381\)[/tex] oranges.
- Since [tex]\(40 \neq 381\)[/tex], this inference is False.
6. Inference 5: The samples show little variation for each fruit.
- The variation within each sample can be calculated by finding the difference between the maximum and minimum preferences in each sample.
- For sample 1 (40, 64, 21): Maximum = 64, Minimum = 21, Variation = [tex]\(64 - 21 = 43\)[/tex]
- For sample 2 (43, 63, 19): Maximum = 63, Minimum = 19, Variation = [tex]\(63 - 19 = 44\)[/tex]
- Since the variations are 43 and 44 for samples 1 and 2, respectively, both are greater than 5. Therefore, the samples do not show little variation. This inference is False.
In conclusion, the true inferences based on the data are:
- About twice as many people prefer apples as prefer bananas.
- Bananas are preferred much less frequently than apples or oranges.
