If Sally gave a survey with five colors listed and ask each person who takes the survey to pick one color. She assumes that the results if given to 24 persons the results would be 1/2 blue, 1/12 red, 1/8 blue, 1/4 green, and 1/24 orange. What number of surveys would best prove her assumtion 120 or 240 explain why
240 is better because you get a result that is closer to the true probability of each color when you take a larger number of samples.
To explain why 240 surveys would be better than 120 to prove Sally's assumption, let's break it down:
Sally assumed that if she gave the survey to 24 people, the results would be:
- 1/2 blue
- 1/12 red
- 1/8 blue
- 1/4 green
- 1/24 orange
To test Sally's assumption, we need to compare the observed results from the surveys to her assumed probabilities. The more surveys we conduct, the better our approximation of the true probabilities for each color.
If we only conduct 120 surveys, it may not be enough to capture the true proportions accurately. The observed values could deviate significantly from the assumed probabilities due to random sampling variability. We might get results like:
- 67 blue
- 10 red
- 15 yellow
- 25 green
- 3 orange
These results do not match Sally's assumed probabilities closely enough to support her assumption. With only 120 surveys, the observed results may still fluctuate quite a bit from the expected values due to random chance.
However, by increasing the number of surveys to 240, we increase our sample size and reduce the impact of random sampling variability. This means the observed results are more likely to align closely with the assumed probabilities if Sally's assumption is correct. With a larger sample size, we might get results like:
- 120 blue
- 20 red
- 30 yellow
- 60 green
- 10 orange
These results appear much closer to the expected probabilities, providing stronger evidence to support Sally's assumption.
In summary, conducting 240 surveys would be a better choice because it helps reduce the influence of random sampling variability and provides a more accurate approximation of the true probabilities for each color.