Question

mai took a survey of students in her class to find out how many hours they spend reading each week. here are some summary statistics for the data that mai gathered: mean: 8.5 hours, standard deviation: 5.3 hours, q1: 5 hours, median: 7 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: Calculate the inter - quartile range (IQR)

The formula for IQR is $IQR = Q3 - Q1$. Given $Q1 = 5$ hours and $Q3=11$ hours, so $IQR=11 - 5=6$ hours.

Step2: Determine the upper - fence for outliers

The upper - fence for outliers is given by the formula $Q3+1.5\times IQR$. Substitute $Q3 = 11$ and $IQR = 6$ into the formula: $11+1.5\times6=11 + 9=20$ hours. Any value greater than 20 hours is an outlier. A number larger than the median (7 hours) and an outlier could be 21 hours. Since $21>20$ (the upper - fence), it is an outlier.

Step3: Determine the lower - fence for outliers

The lower - fence for outliers is given by the formula $Q1 - 1.5\times IQR$. Substitute $Q1 = 5$ and $IQR = 6$ into the formula: $5-1.5\times6=5 - 9=- 4$ hours. Since the number of hours spent reading cannot be negative, the lowest possible non - negative value is 0 hours. All non - negative values greater than or equal to 0 and less than the median (7 hours) are within the range defined by the lower - fence and the median. So there are no outliers below the median.

Answer:

a. 21 hours. Reason: The upper - fence for outliers is $Q3 + 1.5\times IQR=11+1.5\times6 = 20$ hours, and 21>20.
b. No. Reason: The lower - fence for outliers is $Q1-1.5\times IQR=5 - 9=-4$ hours. Since the number of hours cannot be negative, all non - negative values less than the median (7 hours) are within the normal range.