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Answer :
To determine which of the given conditions is not necessary for constructing a confidence interval for the proportion, let's go through each condition step by step:
1. Normal/Large Sample Condition: This condition states that the sample size should be large enough. In this case, the sample size is 50, which is greater than or equal to 30. Therefore, this condition is satisfied.
2. Large Counts Condition (First Statement): This condition ensures that the expected counts in both categories (planning to attend and not planning to attend) are large enough. Here, [tex]\( n \hat{p} = 36 \)[/tex], which is the number of seniors planning to attend the prom. Since 36 is greater than or equal to 10, this condition is satisfied.
3. Random Condition: The sample should be a simple random sample (SRS). In the problem, it is explicitly stated that an SRS of 50 seniors is used. Therefore, this condition is satisfied.
4. 10% Condition: This condition is met if the sample size [tex]\( n \)[/tex] is less than 10% of the population size. The sample size is 50, and the total number of seniors is 750. Since 50 is indeed less than 10% of 750, this condition is satisfied.
5. Large Counts Condition (Second Statement): This condition checks the count of seniors not planning to attend the prom. We calculate [tex]\( n(1-\hat{p}) = 50 - 36 = 14 \)[/tex]. Since 14 is greater than or equal to 10, this condition is satisfied.
Upon reviewing these conditions, it becomes clear that they are all typically required to ensure the validity of a confidence interval; however, the problem statement asks for which condition is not necessary for constructing a confidence interval. Thus, Condition 5 might seem redundant or incorrectly stated compared to the typical conditions, but it's actually still valid.
Therefore, based on what each condition is described as, all these conditions are appropriately necessary, and typically all should be met for a proper confidence interval. The problem likely wants us to identify a condition that does not fit the typical structure of conditions required for constructing a confidence interval, but all listed seem necessary.
Given the earlier conclusion, the supposed answer that is not one of the conditions might have been expected differently, but based on our understanding, all conditions indeed fit typical calculations unless stated otherwise by an examination or source context interpretation differing from standard statistical practice.
1. Normal/Large Sample Condition: This condition states that the sample size should be large enough. In this case, the sample size is 50, which is greater than or equal to 30. Therefore, this condition is satisfied.
2. Large Counts Condition (First Statement): This condition ensures that the expected counts in both categories (planning to attend and not planning to attend) are large enough. Here, [tex]\( n \hat{p} = 36 \)[/tex], which is the number of seniors planning to attend the prom. Since 36 is greater than or equal to 10, this condition is satisfied.
3. Random Condition: The sample should be a simple random sample (SRS). In the problem, it is explicitly stated that an SRS of 50 seniors is used. Therefore, this condition is satisfied.
4. 10% Condition: This condition is met if the sample size [tex]\( n \)[/tex] is less than 10% of the population size. The sample size is 50, and the total number of seniors is 750. Since 50 is indeed less than 10% of 750, this condition is satisfied.
5. Large Counts Condition (Second Statement): This condition checks the count of seniors not planning to attend the prom. We calculate [tex]\( n(1-\hat{p}) = 50 - 36 = 14 \)[/tex]. Since 14 is greater than or equal to 10, this condition is satisfied.
Upon reviewing these conditions, it becomes clear that they are all typically required to ensure the validity of a confidence interval; however, the problem statement asks for which condition is not necessary for constructing a confidence interval. Thus, Condition 5 might seem redundant or incorrectly stated compared to the typical conditions, but it's actually still valid.
Therefore, based on what each condition is described as, all these conditions are appropriately necessary, and typically all should be met for a proper confidence interval. The problem likely wants us to identify a condition that does not fit the typical structure of conditions required for constructing a confidence interval, but all listed seem necessary.
Given the earlier conclusion, the supposed answer that is not one of the conditions might have been expected differently, but based on our understanding, all conditions indeed fit typical calculations unless stated otherwise by an examination or source context interpretation differing from standard statistical practice.
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