Question

3 mai took a survey of students in her class to find out how many hours they spend each week. here are some summary statistics for the data that mai gathered: mean 8.5 hours, median 7 hours, standard deviation 5.3 hours, q1: 5 hours, q3: 11 hours. a. give an example of a number of hours larger than the median which would be an outlier. explain your reasoning. b. are there any outliers below the median? explain your reasoning.

Explanation:

Step1: Recall outlier formula

An outlier is defined as a value less than $Q1 - 1.5\times IQR$ or greater than $Q3+1.5\times IQR$, where $IQR = Q3 - Q1$. First, calculate $IQR$.
$IQR=Q3 - Q1=11 - 5 = 6$

Step2: Calculate upper - bound for non - outliers

Calculate the upper - bound for non - outliers using the formula $Q3+1.5\times IQR$.
$Q3+1.5\times IQR=11+1.5\times6=11 + 9=20$

Step3: Provide an outlier example

A value larger than the median (7 hours) and also an outlier could be 25 hours. Since $25>20$, it is an outlier.

Step4: Calculate lower - bound for non - outliers

Calculate the lower - bound for non - outliers using the formula $Q1 - 1.5\times IQR$.
$Q1 - 1.5\times IQR=5-1.5\times6=5 - 9=- 4$
Since the number of hours cannot be negative, the lowest non - negative value we consider for non - outliers starts from 0. All values between 0 and the median (7 hours) are not outliers. So, there are no outliers below the median.

Answer:

a. An example of a number of hours larger than the median which is an outlier is 25 hours. Reason: $IQR = Q3 - Q1=11 - 5 = 6$, $Q3+1.5\times IQR=11 + 9=20$, and $25>20$.
b. No. Reason: $Q1 - 1.5\times IQR=5 - 9=-4$, and since the number of hours cannot be negative, all non - negative values less than the median (7 hours) are within the non - outlier range.