Question

In a recent study, \(^{24} 2006\) randomly selected US adults (age 18 or older) were asked to give the number of people in the last six months "with whom you discussed matters that are important to you." The average number of close confidants was 2.2 with a standard deviation of 1.4 (a) Find the margin of error for this estimate if we want \(99 \%\) confidence. (b) Find and interpret a \(99 \%\) confidence interval for average number of close confidants.

Short answer

The margin of error with 99% confidence is 0.0734. The 99% confidence interval for the average number of close confidants is \(2.1266, 2.2734\), meaning that we can be 99% confident that the true population mean falls within this range.

Step-by-step solution

Calculate the Standard Error

The standard error (SE) is found by dividing the standard deviation by the square root of the sample size: \nSE = \( \frac{1.4}{\sqrt{2406}} = 0.0285 \)

Find the Z-value for 99% Confidence

A Z table or calculator is usually used for this. The Z value for 99% confidence is \(2.576\).

Calculate the Margin of Error

The margin of error for a sample is calculated by multiplying the Z value by the standard error: \nMargin of Error = \(2.576 * 0.0285 = 0.0734 \)

Calculate the Confidence Interval

The confidence interval is found by taking the mean and adding and subtracting the margin of error. In this case, the mean was 2.2. So the confidence interval is: \n\(2.2 - 0.0734, 2.2 + 0.0734 = (2.1266, 2.2734)\)

Interpret the Results

This means that we can be 99% confident that the population mean number of close confidants falls between 2.1266 and 2.2734.