Answer :
When constructing a confidence interval for a population proportion, checking the Large Counts condition is important for accurate results. Let's break down the solution step-by-step, addressing both parts of the question:
a. Why is it necessary to check this condition?
The Large Counts condition requires that both [tex]\( np \)[/tex] and [tex]\( n(1 - p) \)[/tex] are at least 10. Here's why this is important:
1. Normal Approximation: The condition ensures that the sampling distribution of the sample proportion can be approximated by a normal distribution. This approximation is crucial because many statistical methods, including the construction of confidence intervals, rely on the assumption that the sample distribution is normal.
2. Central Limit Theorem: According to the Central Limit Theorem, as the sample size increases, the distribution of the sample proportion becomes approximately normal, even if the underlying population distribution is not normal. The requirement that both [tex]\( np \)[/tex] and [tex]\( n(1 - p) \)[/tex] are at least 10 helps to ensure that the sample size is large enough for this approximation to hold true.
3. Statistical Calculations: When the sample distribution is approximately normal, we can use the properties of the normal distribution to calculate confidence intervals and conduct hypothesis tests with accuracy. This requires the normal approximation to be valid, which depends on the Large Counts condition being met.
b. What happens to the capture rate if this condition is violated?
If the Large Counts condition is not satisfied, meaning that either [tex]\( np < 10 \)[/tex] or [tex]\( n(1 - p) < 10 \)[/tex], several issues can arise:
1. Inaccuracy in Confidence Levels: If the condition is violated, the confidence interval may not accurately reflect the true variability of the population proportion. The capture rate, or the probability that the interval contains the true population proportion, can become unreliable.
2. Lower Capture Rate: The actual confidence level may be lower than what is stated, leading to a confidence interval that does not accurately achieve the intended confidence level (e.g., 95%).
3. Misleading Conclusions: This can result in misleading conclusions, as the interval may fail to capture the true population parameter at the desired level, leading to potential errors in interpretation and decision-making.
In summary, checking the Large Counts condition is essential for ensuring that the normal approximation of the sampling distribution is valid. Violating this condition can lead to inaccurate confidence intervals and conclusions about the population proportion.
a. Why is it necessary to check this condition?
The Large Counts condition requires that both [tex]\( np \)[/tex] and [tex]\( n(1 - p) \)[/tex] are at least 10. Here's why this is important:
1. Normal Approximation: The condition ensures that the sampling distribution of the sample proportion can be approximated by a normal distribution. This approximation is crucial because many statistical methods, including the construction of confidence intervals, rely on the assumption that the sample distribution is normal.
2. Central Limit Theorem: According to the Central Limit Theorem, as the sample size increases, the distribution of the sample proportion becomes approximately normal, even if the underlying population distribution is not normal. The requirement that both [tex]\( np \)[/tex] and [tex]\( n(1 - p) \)[/tex] are at least 10 helps to ensure that the sample size is large enough for this approximation to hold true.
3. Statistical Calculations: When the sample distribution is approximately normal, we can use the properties of the normal distribution to calculate confidence intervals and conduct hypothesis tests with accuracy. This requires the normal approximation to be valid, which depends on the Large Counts condition being met.
b. What happens to the capture rate if this condition is violated?
If the Large Counts condition is not satisfied, meaning that either [tex]\( np < 10 \)[/tex] or [tex]\( n(1 - p) < 10 \)[/tex], several issues can arise:
1. Inaccuracy in Confidence Levels: If the condition is violated, the confidence interval may not accurately reflect the true variability of the population proportion. The capture rate, or the probability that the interval contains the true population proportion, can become unreliable.
2. Lower Capture Rate: The actual confidence level may be lower than what is stated, leading to a confidence interval that does not accurately achieve the intended confidence level (e.g., 95%).
3. Misleading Conclusions: This can result in misleading conclusions, as the interval may fail to capture the true population parameter at the desired level, leading to potential errors in interpretation and decision-making.
In summary, checking the Large Counts condition is essential for ensuring that the normal approximation of the sampling distribution is valid. Violating this condition can lead to inaccurate confidence intervals and conclusions about the population proportion.
