Question
Algebra 2B Semester Exam Two best friends want to buy a box of colored marbles. Accordingly, 25.6 percent of the marbles in a box are burgundy. Kean buys an extra-large box of these colored marbles. Aykut buys a small box. Which box is more likely to contain 45 percent of the burgundy-colored marbles? (1 point) Both boxes have an equal chance of containing more than 45 percent burgundy marbles because they are both random samples. The extra-large box, as there are more marbles, and thus a greater likelihood of more burgundy marbles. The extra-large box, as there is more variability in the proportion of burgundy marbles among a larger sample size. The small box, as there is more variability in the proportion of burgundy marbles among a smaller sample size
Answers
GPT-4o mini
To determine which box is more likely to contain 45 percent of the burgundy-colored marbles, we need to consider the concept of variability in sample sizes.
1. **Both boxes have an equal chance of containing more than 45 percent burgundy marbles because they are both random samples.** - This statement is incorrect because the sample size affects variability.
2. **The extra-large box, as there are more marbles, and thus a greater likelihood of more burgundy marbles.** - This statement is misleading. While the extra-large box has more marbles, it's important to recognize how the proportion of burgundy marbles behaves with larger sample sizes.
3. **The extra-large box, as there is more variability in the proportion of burgundy marbles among a larger sample size.** - This statement is also incorrect. As sample sizes increase, the variability (or standard error) of the sample proportion decreases, leading to a narrower range of expected proportions.
4. **The small box, as there is more variability in the proportion of burgundy marbles among a smaller sample size.** - This statement is correct. The smaller box will have more variability in its proportion of burgundy marbles, which means it is more likely to deviate from the expected 25.6 percent and could contain a higher percentage, such as 45 percent.
Therefore, the best answer is: **The small box, as there is more variability in the proportion of burgundy marbles among a smaller sample size.**
1. **Both boxes have an equal chance of containing more than 45 percent burgundy marbles because they are both random samples.** - This statement is incorrect because the sample size affects variability.
2. **The extra-large box, as there are more marbles, and thus a greater likelihood of more burgundy marbles.** - This statement is misleading. While the extra-large box has more marbles, it's important to recognize how the proportion of burgundy marbles behaves with larger sample sizes.
3. **The extra-large box, as there is more variability in the proportion of burgundy marbles among a larger sample size.** - This statement is also incorrect. As sample sizes increase, the variability (or standard error) of the sample proportion decreases, leading to a narrower range of expected proportions.
4. **The small box, as there is more variability in the proportion of burgundy marbles among a smaller sample size.** - This statement is correct. The smaller box will have more variability in its proportion of burgundy marbles, which means it is more likely to deviate from the expected 25.6 percent and could contain a higher percentage, such as 45 percent.
Therefore, the best answer is: **The small box, as there is more variability in the proportion of burgundy marbles among a smaller sample size.**