Math Problem Statement

Select all of the number of hours larger than the median which would be an outlier, given the statistics: mean = 8.5 hours, MAD = 5.3 hours, median = 7 hours, Q1 = 5 hours, Q3 = 11 hours.

Solution

To determine which values are outliers larger than the median in this dataset, we follow these steps:


1. Outlier Boundaries

The formula for identifying outliers is:

Upper Bound=Q3+1.5×IQR\text{Upper Bound} = Q3 + 1.5 \times \text{IQR}

Where:

  • Q3=11Q3 = 11
  • Q1=5Q1 = 5
  • IQR=Q3Q1\text{IQR} = Q3 - Q1

Calculating the IQR: IQR=115=6\text{IQR} = 11 - 5 = 6

Upper Bound: Upper Bound=11+1.5×6=11+9=20\text{Upper Bound} = 11 + 1.5 \times 6 = 11 + 9 = 20

Thus, any value greater than 20 is considered an outlier.


2. Checking Given Values

The options are:

  • 20: This is not an outlier since it equals the upper bound.
  • 21: This is an outlier since it is greater than 20.
  • 25: This is an outlier since it is greater than 20.
  • 15: This is not an outlier since it is less than the upper bound.

3. Correct Answers

The numbers larger than the median (77) and classified as outliers are:

  • 21
  • 25

Final Analysis

The correct selections should include 21 and 25 only.

Would you like more details on how I calculated the boundaries? Let me know!


Related Questions:

  1. How is the IQR (Interquartile Range) used to determine outliers?
  2. Why is the threshold 1.5×IQR1.5 \times \text{IQR} used in statistics?
  3. What happens if the data has extreme outliers on both ends?
  4. How does the mean differ from the median in skewed datasets?
  5. How would you identify outliers in a box plot?

Tip:

Always verify both lower and upper outlier bounds when analyzing a dataset.

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Math Problem Analysis

Mathematical Concepts

Outliers
Interquartile Range (IQR)
Median
Upper Bound for Outliers

Formulas

Upper Bound = Q3 + 1.5 * IQR
IQR = Q3 - Q1

Theorems

Outlier Detection Rule based on IQR

Suitable Grade Level

Grades 8-10