Question

According to an AP-Ipsos poll (June 15,2005 ), \(42 \%\) of 1001 randomly selected adult Americans made plans in May 2005 based on a weather report that turned out to be wrong. a. Construct and interpret a \(99 \%\) confidence interval for the proportion of Americans who made plans in May 2005 based on an incorrect weather report. b. Do you think it is reasonable to generalize this estimate to other months of the year? Explain.

Short answer

a. The 99% confidence interval for the proportion of Americans who made plans based on an incorrect weather report in May 2005 would be (lower limit, upper limit) calculated in step 4. b. Whether or not it's reasonable to extend this to other months depends on your belief about seasonal effects. If you believe people's behavior towards weather reports doesn't change drastically per season, it might be okay to extend.

Step-by-step solution

Use the formula for the confidence interval for proportions

The formula to calculate the confidence interval for proportions is \(p ± Z*(√((p(1 - p))/n))\) where \(p\) is the sample proportion (0.42), \(n\) is the sample size (1001), and \(Z\) is the Z-score corresponding to the desired confidence level (for 99%, Z = 2.576).

Find the standard error

Substitute the given data into the formula \(√(p(1 - p))/n\). This means, \(√((0.42(1 - 0.42))/1001)\). The result will be used in the next step

Compute the margin of error

Multiply the Z-score by the standard error calculated in step 2 to find the margin of error. \(Z*standard\ error = 2.576 * standard\ error\) computed in step 2

Find the confidence interval

Subtract the margin of error from the sample proportion for the lower limit and add the margin of error to the sample proportion for the upper limit. This gives the 99% confidence interval for this case

Answer part b

Deciding if this could extend to future months depends on whether you believe the behavior of people in May (for weather reports) is drastically different from other months. If it is believed there's no significant seasonal effect, the intervals could extend. Otherwise, they might not.