Shania is buying a car. She has narrowed her choice down to three cars: a sedan, a hatchback, and a wagon. She uses a spinner divided into three colored sections to decide which car to buy. She first spins the spinner 75 times to check the frequencies that it produces. Her results are shown in the table.
| Car | Spinner Color | Times Result Occurred |
| --------- | ------------- | ----------------------- |
| sedan | red | 11 |
| hatchback | blue | 22 |
| wagon | green | 42 |
Which statement is true based on the results in the table?
A. All the outcomes are equally likely because their relative frequencies are similar.
B. All the outcomes are equally likely because the relative frequencies of landing on at least two colors are similar.
C. All the outcomes are not equally likely because there is a variation in their relative frequencies.
D. All the outcomes are not equally likely because the spinner color is not a valid representation of the car color.
E. All the outcomes are not equally likely because the relative frequency of landing on red is the least.
Answer (1)
Calculate relative frequencies: red ≈ 0.1467 , blue ≈ 0.2933 , green = 0.56 .
Compare the relative frequencies.
Outcomes are not equally likely due to variations in relative frequencies.
The correct statement is C: All the outcomes are not equally likely because there is a variation in their relative frequencies.
Explanation
Analyze the problem We are given the number of times each color occurred when spinning a spinner 75 times. We need to determine which statement is true based on these results. The key is to analyze the relative frequencies of each color.
Calculate relative frequencies First, let's calculate the relative frequencies for each color:
Red: 75 11 ≈ 0.1467
Blue: 75 22 ≈ 0.2933
Green: 75 42 = 0.56
Evaluate each statement Now, let's evaluate the given statements:
A. All the outcomes are equally likely because their relative frequencies are similar. This is incorrect because the relative frequencies are quite different (0.1467, 0.2933, and 0.56).
B. All the outcomes are equally likely because the relative frequencies of landing on at least two colors are similar. This statement is confusing and doesn't make sense in the context of equally likely outcomes.
C. All the outcomes are not equally likely because there is a variation in their relative frequencies. This is correct because the relative frequencies are different, indicating that the outcomes are not equally likely.
D. All the outcomes are not equally likely because the spinner color is not a well representation of the car color. This statement is irrelevant to the data provided; the spinner's representation of car color doesn't affect the likelihood of each outcome.
E. All the outcomes are not equally likely because the relative frequency of landing on red is the least. While this is true, option C is a more comprehensive explanation.
Conclusion Based on the analysis, the correct statement is C: All the outcomes are not equally likely because there is a variation in their relative frequencies.
Examples
Understanding relative frequencies can help in various real-life scenarios. For example, in marketing, businesses track how often customers choose certain products to optimize their inventory and marketing strategies. Similarly, in weather forecasting, meteorologists analyze historical data to determine the likelihood of different weather conditions occurring, helping them make more accurate predictions. In sports, coaches analyze player performance data to understand the frequency of successful plays and adjust their strategies accordingly.
