Question

In the article "Fluoridation Brushed Off by Utah" (Associated Press, August 24,1998 ), it was reported that a small but vocal minority in Utah has been successful in keeping fluoride out of Utah water supplies despite evidence that fluoridation reduces tooth decay and despite the fact that a clear majority of Utah residents favor fluoridation. To support this statement, the article included the result of a survey of Utah residents that found \(65 \%\) to be in favor of fluoridation. Suppose that this result was based on a random sample of 150 Utah residents. Construct and interpret a \(90 \%\) confidence interval for \(p,\) the proportion of all Utah residents who favor fluoridation. Is this interval consistent with the statement that fluoridation is favored by a clear majority of residents?

Short answer

The 90% confidence interval is calculated to estimate the true proportions. If the lower limit of the interval is greater than 0.5, it indicates a clear majority favoring fluoridation. The specific numerical values will be determined by the calculations.

Step-by-step solution

Identify Given Information

We know that the sample size (n) is 150 and the sample proportion (\( \hat{p} \)) is 0.65 or 65%.

Calculate Standard Error

Next, we need to calculate the standard error (SE). This can be done using the formula for standard error of a proportion which is \( SE = \sqrt{ \frac{\hat{p}*(1-\hat{p})}{n} }\). Therefore, \( SE = \sqrt{ \frac{0.65*(1 - 0.65)}{150} } \).

Construct the Confidence Interval

A 90% confidence interval for the population proportion can be constructed using the formula \( \hat{p} \pm Z*SE \), where Z is a value from the standard normal distribution related to the desired confidence level. For a 90% confidence interval, Z is approximately 1.645. Substitute the previously calculated values into the formula.

Interpret the Confidence Interval

The constructed interval can be interpreted as we are 90% confident that the true proportion of Utah residents who favor fluoridation is within this interval.

Check if Majority Favors Fluoridation

To check if a clear majority of the residents favor fluoridation, we need to see if the lower limit of the confidence interval is greater than 0.5 (or 50%). If that's the case, then it supports the statement that a clear majority favors fluoridation.