If a 95% confidence Interval for prevalence of cancer smokers aged > 65 years is 56% to 76%, the chance that the prevalance could be less than 56% is ?
**Core Concept:** Confidence Interval (CI) is a statistical tool used to express the precision of a sample-based estimate. In this case, the confidence interval is given for the prevalence of cancer among smokers aged >65 years.
**Why the Correct Answer is Right:**
The correct answer (C) is calculated using the formula for the lower limit of a 95% confidence interval:
**Correct Answer Formula:** Lower Limit = x̄ - (z * (se)^2)
Where:
x̄ is the sample mean (prevalence),
z is the z-score (1.96 for a 95% confidence interval),
se is the sample standard error (standard deviation divided by √n, where n is the sample size).
**Why Each Wrong Option is Incorrect:**
A. This option is not calculated using the correct formula for the lower limit of a 95% confidence interval.
B. This option is not calculated using the correct formula for the lower limit of a 95% confidence interval.
D. This option is not calculated using the correct formula for the lower limit of a 95% confidence interval.
**Why Each Wrong Option is Incorrect:**
A. 95% confidence interval is not a probability, so stating "probability = 0.025" is incorrect.
B. 95% confidence interval is not a probability, so stating "probability = 0.025" is incorrect.
D. 95% confidence interval is not a probability, so stating "probability = 0.025" is incorrect.
**Why Each Wrong Option is Incorrect:**
A. The probability of the prevalence being less than the lower limit is not calculated using the formula for the probability of the lower limit.
B. The probability of the prevalence being less than the lower limit is not calculated using the formula for the probability of the lower limit.
D. The probability of the prevalence being less than the lower limit is not calculated using the formula for the probability of the lower limit.
**Clinical Pearl:**
When interpreting a confidence interval, it is important to understand the relationship between the confidence interval and the probability of the population parameter falling within the interval. A 95% confidence interval indicates that the true prevalence of the population (the parameter) is likely to fall within the calculated interval 95% of the time if the sample is representative of the population. The probability of the prevalence being less than the lower limit is calculated by finding the probability that the lower limit is less than the true prevalence.
**Why the Correct Answer is 0.025:**
The correct answer (C) is calculated by finding the probability that the lower limit (0.025) is less than the true prevalence.
**Why Each Wrong Option is Incorrect:**
Option A (0.0025) is too small and does not represent the probability of the lower limit being less than the true prevalence.
Option B (0.05) is not related to the probability of the lower limit being less than the true prevalence.
Option D