1. A bag contains solid-color marbles, of which some are red and the rest are blue. A random sample of 16 marbles is selected from the bag, and 3 of the 16 marbles are red. If the bag contains 192 marbles, what is the most reasonable estimate for the number of red marbles in the bag?
Answer: D
The most reasonable estimate for the number of red marbles in the bag is 36.
To estimate the number of red marbles in the bag, we can use the proportion of red marbles in the sample to project the total. With 3 out of 16 marbles being red, the estimated number of red marbles in a total of 192 marbles is approximately 36.
A) 12
Choosing 12 would imply a much lower proportion of red marbles than observed in the sample. Given that 3 out of 16 marbles are red, this would suggest a significant underestimation of the total number of red marbles in the bag.
B) 18
An estimate of 18 red marbles also fails to align with the proportion found in the sample. This would indicate roughly 9% of the total marbles are red, which is inconsistent with the observed ratio of approximately 18.75% (3 out of 16).
C) 31
Estimating 31 red marbles suggests a proportion of about 16.15%. While this is closer to the observed ratio, it still does not accurately reflect the sample's findings and remains below the rounded estimate based on the sample size.
D) 36
Estimating 36 red marbles corresponds closely to the observed ratio of red marbles from the sample. It represents 18.75% of the total 192 marbles, which aligns well with the ratio seen in the sample of 16 marbles, making it the most reasonable estimate.
Conclusion
The correct estimate of 36 red marbles accurately reflects the proportion observed in the sample taken from the bag. Other options significantly underestimate the number of red marbles, not aligning with the sample's ratio and thereby failing to provide a plausible estimate. Thus, 36 is the only option consistent with the data presented.