Question

In Exercises 3.51 to 3.56 , information about a sample is given. Assuming that the sampling distribution is symmetric and bell-shaped, use the information to give a \(95 \%\) confidence interval, and indicate the parameter being estimated. $$ r=0.34 \text { and the standard error is } 0.02 \text { . } $$

Short answer

The \(95\%\) confidence interval for the population correlation coefficient 'r' is \(0.302, 0.378\).

Step-by-step solution

Find the Z-score

First, we need to find the Z-score that corresponds to a \(95\%\) confidence level. Since we want the middle \(95\%\) , we are left with \(5\%\) in the two tails of the Normal Distribution. Half of \(5\%\) is \(2.5\%\) in each tail. So, we look up \(2.5\%\) in the Z-table or use a calculator to find the Z-score. The Z-score for \(95\%\) confidence is roughly \(\pm 1.96\).

Insert into the formula

Now we have all the values to insert into our confidence interval formula \(X \pm Z * SE\). Here, \(X = 0.34\), which is the given sample statistic, \(Z = 1.96\), and \(SE = 0.02\). Substitute these values into the formula to get \(0.34 \pm 1.96 * 0.02\).

Calculate the confidence interval

Calculate the upper and lower limit of the confidence interval by performing the operations. The lower limit is \(0.34 - 1.96 * 0.02 = 0.302\), and the upper limit is \(0.34 + 1.96 * 0.02 = 0.378\).

Interpret the result

The \(95\%\) confidence interval is \(0.302, 0.378\). This means that we're \(95\%\) confident that the actual population parameter lies between these two numbers. The parameter being estimated here is 'r', the population correlation coefficient.