Question

What Proportion Believe in One True Love? In Data 2.1 on page 48 , we describe a study in which a random sample of 2625 US adults were asked whether they agree or disagree that there is "only one true love for each person." The study tells us that 735 of those polled said they agree with the statement. The standard error for this sample proportion is \(0.009 .\) Define the parameter being estimated, give the best estimate, the margin of error, and find and interpret a \(95 \%\) confidence interval.

Short answer

The parameter being estimated is the proportion of all US adults who believe in 'only one true love for each person'. The best estimate of this parameter is 0.28 or 28%. The margin of error for this estimate is 0.018 or 1.8%. Thus, a 95% confidence interval for the proportion of all US adults who agree with the statement is from 26.2% to 29.8%.

Step-by-step solution

Defining the Parameter

The parameter being estimated in this problem is the proportion of all US adults who believe that 'there is only one true love for each person'.

Calculating the Best Estimate

The best estimate of the parameter is the sample proportion, \(p\). We can calculate this by dividing the number of people who agreed with the statement (735) by the total number of people surveyed (2625). So, \(p = 735 / 2625 = 0.28\). This suggests that 28% of the surveyed population agree with the statement.

Computing the Margin of Error

The margin of error can be calculated using the formula for a 95% confidence interval, which is ± 1.96 times the standard error. Given that the standard error, \(SE\), is 0.009, our margin of error is \(1.96*SE = 1.96*0.009 = 0.018\). This gives us a margin of error of 0.018 or 1.8%.

Constructing and Interpreting a 95% Confidence Interval

A 95% confidence interval is constructed around the best estimate and extends the margin of error on either side. The lower limit of the confidence interval will be \(p - error = 0.28 - 0.018 = 0.262\) and the upper limit will be \(p + error = 0.28 + 0.018 = 0.298\). So, we are 95% confident that the true proportion of all US adults who agree with the statement falls between 26.2% and 29.8%.