Paris used a simulation to take two random samples of fish in a pond. Her sample size was 30, and the table shows the frequency of each fish in the sample. Paris used the two samples to predict that the average number of trout will be 6, the average number of catfish will be 5, and the average number of bass will be 18.
| Fish Type | Frequency Sample 1 | Frequency Sample 2 |
|---|---|---|
| Brook Trout | 12 | 1 |
| Catfish | 5 | 6 |
| Smallmouth Bass | 13 | 23 |
Is Paris correct in her predictions? Check all that apply.
A. Yes, her samples represent the population.
B. No, Paris should have used only the results of the first sample.
C. No, Paris should have used only the results of the second sample.
D. No, the two samples do not have similar values, so more samples are needed.
E. No, the samples are not representative of the population.
Answer (1)
Calculate the average frequency of each fish type across the two samples: Trout average is 2 12 + 1 = 6.5 , Catfish average is 2 5 + 6 = 5.5 , and Bass average is 2 13 + 23 = 18 .
Compare the calculated averages with Paris's predictions: Trout (6.5 vs. 6), Catfish (5.5 vs. 5), and Bass (18 vs. 18).
Notice that the two samples do not have similar values, indicating that more samples are needed for a reliable estimate.
Conclude that the samples are not fully representative of the population, as the calculated averages differ from Paris's predictions. Therefore, the correct answers are: No, the two samples do not have similar values, so more samples are needed. No, the samples are not representative of the population.
Explanation
Analyze the data and predictions First, let's analyze the data. Paris took two samples of fish from a pond and recorded the frequency of each type of fish in each sample. She then made predictions about the average number of each fish type. We need to check if her predictions are accurate based on the given data.
Calculate average frequencies Now, let's calculate the average frequency for each fish type across the two samples.
Brook Trout: The frequencies are 12 and 1. The average is calculated as follows: 2 12 + 1 = 2 13 = 6.5
Catfish: The frequencies are 5 and 6. The average is calculated as follows: 2 5 + 6 = 2 11 = 5.5
Smallmouth Bass: The frequencies are 13 and 23. The average is calculated as follows: 2 13 + 23 = 2 36 = 18
Compare calculated averages with predictions Now, let's compare the calculated averages with Paris's predictions:
Paris predicted an average of 6 trout, but the calculated average is 6.5. These values are close, but not exactly the same.
Paris predicted an average of 5 catfish, but the calculated average is 5.5. These values are close, but not exactly the same.
Paris predicted an average of 18 bass, and the calculated average is 18. These values are the same.
Evaluate the statements Based on our calculations and comparisons, let's evaluate the given statements:
"Yes, her samples represent the population." This statement is not entirely accurate since the trout and catfish averages are not exactly as predicted, suggesting the samples might not perfectly represent the population.
"No, Paris should have used only the results of the first sample." This is incorrect because using both samples generally provides a more accurate representation of the population than using a single sample.
"No, Paris should have used only the results of the second sample." This is also incorrect for the same reason as above.
"No, the two samples do not have similar values, so more samples are needed." The values for trout vary significantly (12 vs. 1), and while the catfish values are closer (5 vs. 6), the bass values also have a notable difference (13 vs. 23). This suggests that more samples could provide a more reliable estimate.
"No, the samples are not representative of the population." This is a reasonable conclusion, as the averages calculated from the samples differ from Paris's predictions, indicating that the samples may not fully represent the population.
Final Answer Therefore, the correct statements are:
No, the two samples do not have similar values, so more samples are needed.
No, the samples are not representative of the population.
Examples
Imagine you're trying to estimate the average height of students in a school. You take two small samples of students and measure their heights. If the average heights in the two samples are quite different, it suggests that you need to take more samples to get a more accurate estimate of the average height of all students in the school. This is similar to Paris needing more samples of fish to get a more accurate estimate of the average number of each fish type in the pond. Using more samples helps to reduce the impact of random variations and provides a better representation of the entire population.
