An online site presented this question, 'Would the recent norovirus outbreak deter you from taking a cruise?' Among 34,020 people who responded, 64% answered 'yes'. Use the sample data to construct a 90% confidence interval estimate for the proportion of
To construct a confidence interval estimate for the proportion of people who would be deterred from taking a cruise due to the recent norovirus outbreak, we can follow these steps:
1. Identify the sample proportion: In this case, the sample proportion is given as 64% or 0.64.
2. Determine the critical value: For a 90% confidence interval, we need to find the critical value from the standard normal distribution. The critical value for a 90% confidence interval is approximately 1.645.
3. Calculate the standard error (SE): The standard error is a measure of how much the sample proportion varies from the true population proportion. It can be calculated using the formula:
SE = √[(p * (1-p)) / n]
where p is the sample proportion and n is the sample size.
In this case, p = 0.64 and n = 34,020. Plug in these values:
SE = √[(0.64 * (1-0.64)) / 34,020]
4. Calculate the margin of error: The margin of error represents the range within which we can be confident the true population proportion lies. It can be calculated by multiplying the standard error by the critical value:
Margin of Error = critical value * SE
In this case, the critical value (from step 2) is 1.645. Plug in the standard error (from step 3):
Margin of Error = 1.645 * (standard error calculated in step 3)
5. Calculate the confidence interval: To obtain the confidence interval, we calculate both the upper and lower bounds:
Lower bound = p - margin of error
Upper bound = p + margin of error
Substitute the equations from step 4 into these formulas:
Lower bound = 0.64 - (1.645 * (standard error calculated in step 3))
Upper bound = 0.64 + (1.645 * (standard error calculated in step 3))
Calculate the values to get the confidence interval.
Using these steps and the given sample data, the 90% confidence interval estimate for the proportion of people who would be deterred from taking a cruise due to the recent norovirus outbreak would be determined.