Question

a sample data set has a mean of 69 and a standard deviation of 12. determine whether each of the following sample measurements are outliers. a. 107 b. 75 c. 68 d. 63 a. the measurement 107 is an outlier is not an outlier

Explanation:

Step1: Recall outlier rule

Outliers are typically values more than 1.5 * IQR (Inter - Quartile Range) away from Q1 or Q3. In a normal - like distribution, we can also use the z - score method. A z - score of \(|z|> 3\) is often considered an outlier. The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step2: Calculate z - score for \(x = 107\)

Given \(\mu = 69\) and \(\sigma=12\), then \(z=\frac{107 - 69}{12}=\frac{38}{12}\approx3.17\).

Step3: Determine if it's an outlier

Since \(|z|\approx3.17>3\), 107 is an outlier.

Step4: Calculate z - score for \(x = 75\)

\(z=\frac{75 - 69}{12}=\frac{6}{12}=0.5\). Since \(|z| = 0.5<3\), 75 is not an outlier.

Step5: Calculate z - score for \(x = 68\)

\(z=\frac{68 - 69}{12}=-\frac{1}{12}\approx - 0.08\). Since \(|z|\approx0.08<3\), 68 is not an outlier.

Step6: Calculate z - score for \(x = 63\)

\(z=\frac{63 - 69}{12}=\frac{-6}{12}=-0.5\). Since \(|z| = 0.5<3\), 63 is not an outlier.

Answer:

a. is an outlier
b. is not an outlier
c. is not an outlier
d. is not an outlier