An online site presented this question, 'Would the recent norovirus outbreak deter you from taking a cruise?' Among 34,327 people who responded, 66% answered 'yes'. Use the sample data to construct a 95% confidence interval estimate for the proportion of the population of all people who would respond 'yes' to that question. Does the
confidence interval provide a good estimate or the population proportion?
To construct a confidence interval estimate for the proportion of the population who would respond 'yes' to the question, we can use the sample data and the formula for calculating confidence intervals for proportions.
First, we need to calculate the standard error of the proportion:
SE = sqrt((p * q) / n)
where
- p is the sample proportion (66% or 0.66),
- q is the complement of the sample proportion (1 - p), and
- n is the sample size (34,327).
Substituting the values into the formula:
SE = sqrt((0.66 * (1-0.66)) / 34,327) ≈ 0.0022
Next, to find the margin of error, we can multiply the standard error by the z-score associated with our desired confidence level. For a 95% confidence level, the z-score is approximately 1.96.
Margin of Error = z * SE ≈ 1.96 * 0.0022 ≈ 0.0043
Finally, we can calculate the confidence interval by subtracting and adding the margin of error to the sample proportion:
Lower bound = p - MOE = 0.66 - 0.0043 ≈ 0.6557
Upper bound = p + MOE = 0.66 + 0.0043 ≈ 0.6643
Therefore, the 95% confidence interval estimate for the proportion of the population who would respond 'yes' to the question is approximately 0.6557 to 0.6643.
Now, regarding whether the confidence interval provides a good estimate for the population proportion, we can say that it is a reasonable estimate within the given confidence level. The 95% confidence interval means that if we were to repeat this survey multiple times and construct confidence intervals, approximately 95% of those intervals would contain the true population proportion. However, the confidence interval is not definitive and does not guarantee 100% accuracy. It provides a range of values within which the true population proportion is likely to be found.