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Answer :
To construct a confidence interval for the proportion of seniors planning to attend the prom, several conditions must be met. These conditions help ensure that the statistical methods used will give reliable results. Let's review each condition to determine which is not necessary based on the given information:
1. Random Condition: The sample must be a simple random sample (SRS). This is important because it helps ensure that the sample is representative of the entire population.
2. Large Counts Condition:
- We check that both [tex]\( n \hat{p} \)[/tex] (the expected number of successes) and [tex]\( n(1 - \hat{p}) \)[/tex] (the expected number of failures) are at least 10. This ensures that the sample size is large enough for the normal approximation to be valid.
- [tex]\( n \hat{p} = 36 \)[/tex], which is greater than or equal to 10.
- [tex]\( n(1 - \hat{p}) = 14 \)[/tex], which is also greater than or equal to 10.
- Thus, the Large Counts condition is satisfied.
3. Normal/Large Sample Condition: This often refers to having a large enough sample size, typically at least 30, so the distribution of the sample proportion is approximately normal. Here, [tex]\( n = 50 \)[/tex], which satisfies this condition since 50 is greater than or equal to 30.
4. 10% Condition: This condition states that the sample size should be less than 10% of the population size. This condition ensures that the independence assumption of the individual sample values is reasonable. Assuming the total number of seniors is at least 1000, the sample size of 50 would be less than 10% of the population.
Based on the review of these conditions, we see that all conditions mentioned are typically applied in constructing a confidence interval for proportion. However, the condition labeled "Normal/Large Sample: The sample size is large [tex]\( n=50 \geq 30 \)[/tex]" is often included in the context of meeting the Large Counts Condition, and thus could be considered somewhat redundant in this specific context. Nonetheless, it's not strictly incorrect but is a repetition of checking that the sample size supports a normal approximation.
Therefore, the condition that might not need to be stated separately from Large Counts is the "Normal/Large Sample" condition, as it overlaps with ensuring the normality through the Large Counts check when using the normal approximation for the sample proportion.
1. Random Condition: The sample must be a simple random sample (SRS). This is important because it helps ensure that the sample is representative of the entire population.
2. Large Counts Condition:
- We check that both [tex]\( n \hat{p} \)[/tex] (the expected number of successes) and [tex]\( n(1 - \hat{p}) \)[/tex] (the expected number of failures) are at least 10. This ensures that the sample size is large enough for the normal approximation to be valid.
- [tex]\( n \hat{p} = 36 \)[/tex], which is greater than or equal to 10.
- [tex]\( n(1 - \hat{p}) = 14 \)[/tex], which is also greater than or equal to 10.
- Thus, the Large Counts condition is satisfied.
3. Normal/Large Sample Condition: This often refers to having a large enough sample size, typically at least 30, so the distribution of the sample proportion is approximately normal. Here, [tex]\( n = 50 \)[/tex], which satisfies this condition since 50 is greater than or equal to 30.
4. 10% Condition: This condition states that the sample size should be less than 10% of the population size. This condition ensures that the independence assumption of the individual sample values is reasonable. Assuming the total number of seniors is at least 1000, the sample size of 50 would be less than 10% of the population.
Based on the review of these conditions, we see that all conditions mentioned are typically applied in constructing a confidence interval for proportion. However, the condition labeled "Normal/Large Sample: The sample size is large [tex]\( n=50 \geq 30 \)[/tex]" is often included in the context of meeting the Large Counts Condition, and thus could be considered somewhat redundant in this specific context. Nonetheless, it's not strictly incorrect but is a repetition of checking that the sample size supports a normal approximation.
Therefore, the condition that might not need to be stated separately from Large Counts is the "Normal/Large Sample" condition, as it overlaps with ensuring the normality through the Large Counts check when using the normal approximation for the sample proportion.
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