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Answer :
To solve this problem, we need to check whether specific conditions for inference, which are mathematical prerequisites for constructing a confidence interval, are satisfied. These conditions ensure that our statistical conclusions are valid. Here's how we evaluate each condition:
1. Random Condition:
The students were assigned to the plans randomly by flipping a coin, which means this condition is satisfied as the students were not chosen by some bias but rather by chance.
2. 10% Condition:
This condition checks if the sample size is less than 10% of the population. However, in this problem, we are not given information about a larger population beyond the classroom. When using the 10% condition, the concern is usually more applicable to scenarios involving samples from very large populations. In this problem, the specific classroom is the entire group of interest, which makes the application of the 10% condition tricky. Without further information about a larger population than the classroom size, it might be assumed unmet.
3. Large Counts Condition:
For this condition to be met, the number of successes and failures in each group should be at least 10. Let's check:
- Plan A: 23 students successfully completed the homework, and [tex]\(34 - 23 = 11\)[/tex] students did not. Both numbers are greater than or equal to 10.
- Plan B: 20 students successfully completed the homework, and [tex]\(26 - 20 = 6\)[/tex] students did not. Here, while the successes meet the criterion, the failures do not.
Since Plan B does not satisfy the Large Counts condition (as it does not have at least 10 failures), the Large Counts condition is not met.
Upon evaluating these conditions, we identify which ones are not satisfied:
- Although the Random condition is met, both the 10% condition and Large Counts are questionable, with the Large Counts condition notably unmet for Plan B. However, considering each condition carefully without additional population context, it is primarily the Large Counts condition that is not met.
Therefore, based on this examination, the answer is:
C) The Large Counts condition is the only condition that is not met.
1. Random Condition:
The students were assigned to the plans randomly by flipping a coin, which means this condition is satisfied as the students were not chosen by some bias but rather by chance.
2. 10% Condition:
This condition checks if the sample size is less than 10% of the population. However, in this problem, we are not given information about a larger population beyond the classroom. When using the 10% condition, the concern is usually more applicable to scenarios involving samples from very large populations. In this problem, the specific classroom is the entire group of interest, which makes the application of the 10% condition tricky. Without further information about a larger population than the classroom size, it might be assumed unmet.
3. Large Counts Condition:
For this condition to be met, the number of successes and failures in each group should be at least 10. Let's check:
- Plan A: 23 students successfully completed the homework, and [tex]\(34 - 23 = 11\)[/tex] students did not. Both numbers are greater than or equal to 10.
- Plan B: 20 students successfully completed the homework, and [tex]\(26 - 20 = 6\)[/tex] students did not. Here, while the successes meet the criterion, the failures do not.
Since Plan B does not satisfy the Large Counts condition (as it does not have at least 10 failures), the Large Counts condition is not met.
Upon evaluating these conditions, we identify which ones are not satisfied:
- Although the Random condition is met, both the 10% condition and Large Counts are questionable, with the Large Counts condition notably unmet for Plan B. However, considering each condition carefully without additional population context, it is primarily the Large Counts condition that is not met.
Therefore, based on this examination, the answer is:
C) The Large Counts condition is the only condition that is not met.
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