A survey was conducted asking people about their favorite fruit. Which inferences about the population are true based on the data? Check all that apply.
| | Apple | Orange | Banana |
| :------ | :---- | :----- | :----- |
| Sample 1 | 40 | 64 | 21 |
| Sample 2 | 43 | 63 | 19 |
A. About twice as many people prefer apples as prefer bananas.
B. Bananas are preferred much less frequently than apples or oranges.
C. Exactly twice as many people prefer oranges as prefer apples.
D. Exactly three times as many people prefer bananas as prefer oranges.
E. The samples show little variation for each fruit.
Answer (2)
Calculate the average preference for each fruit: Apple (41.5), Orange (63.5), Banana (20).
Evaluate if 'About twice as many people prefer apples as prefer bananas': 41.5 ≈ 2 × 20 is true.
Evaluate if 'Bananas are preferred much less frequently than apples or oranges': 20 << 41.5 and 20 << 63.5 is true.
Evaluate if 'The samples show little variation for each fruit' by checking the ranges (Apple: 3, Orange: 1, Banana: 2) are small relative to the averages.
The true inferences are:
About twice as many people prefer apples as prefer bananas.
Bananas are preferred much less frequently than apples or oranges.
The samples show little variation for each fruit.
Explanation
Analyze the problem and data We are given a table with survey data about fruit preferences. Our goal is to evaluate several inferences based on this data. The table provides the number of people who prefer apples, oranges, and bananas in two different samples. We will calculate the average preference for each fruit and the range of preferences across the samples to assess the validity of the given inferences.
Calculate averages and ranges First, let's calculate the average number of people who prefer each fruit across the two samples:
Average number of people who prefer apples: 2 40 + 43 = 2 83 = 41.5
Average number of people who prefer oranges: 2 64 + 63 = 2 127 = 63.5
Average number of people who prefer bananas: 2 21 + 19 = 2 40 = 20
Next, let's calculate the range (maximum - minimum) for each fruit across the two samples:
Range for apples: 43 − 40 = 3 Range for oranges: 64 − 63 = 1 Range for bananas: 21 − 19 = 2
Evaluate each inference Now, let's evaluate each inference:
About twice as many people prefer apples as prefer bananas. We need to check if 41.5 ≈ 2 × 20 .
2 × 20 = 40 . Since 41.5 is close to 40 , this inference is likely true.
Bananas are preferred much less frequently than apples or oranges. We need to check if 20 << 41.5 and 20 << 63.5 .
Since 20 is significantly smaller than both 41.5 and 63.5 , this inference is likely true.
Exactly twice as many people prefer oranges as prefer apples. We need to check if 63.5 = 2 × 41.5 .
2 × 41.5 = 83 . Since 63.5 is not equal to 83 , this inference is false.
Exactly three times as many people prefer bananas as prefer oranges. We need to check if 20 = 3 × 63.5 .
3 × 63.5 = 190.5 . Since 20 is not equal to 190.5 , this inference is false.
The samples show little variation for each fruit. We look at the ranges we calculated: Apple (3), Orange (1), Banana (2). These ranges are small relative to the average preferences (41.5, 63.5, 20), so this inference is likely true.
Determine true inferences Based on our analysis, the true inferences are:
About twice as many people prefer apples as prefer bananas.
Bananas are preferred much less frequently than apples or oranges.
The samples show little variation for each fruit.
Final Answer Therefore, the inferences that are true 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.
The samples show little variation for each fruit.
Examples
Understanding preferences in surveys is crucial for businesses. For example, a fruit juice company might conduct a similar survey to decide which new flavor to launch. By analyzing the survey data, they can identify which fruits are most popular and ensure they invest in a product that appeals to a large segment of the population, maximizing their potential for success.
The true inferences from the survey data are: about twice as many people prefer apples as bananas, bananas are preferred much less than apples or oranges, and the samples show little variation for each fruit.
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