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Chapter 3: Problem 31

Data on weekday exercise time for 20 males, consistent with summary quantities given in the paper "An Ecological Momentary Assessment of the Physical Activity and Sedentary Behaviour Patterns of University Students" (Health Education Journal [2010]\(: 116-125),\) are shown below. Calculate and interpret the values of the median and interquartile range. $$ \begin{array}{rrrrrrrrr} 43.5 & 91.5 & 7.5 & 0.0 & 0.0 & 28.5 & 199.5 & 57.0 & 142.5 \\ 8.0 & 9.0 & 36.0 & 0.0 & 78.0 & 34.5 & 0.0 & 57.0 & 151.5 \\ 8.0 & 0.0 & & & & & & & \end{array} $$

Short Answer

Expert verified
The median weekday exercise time for the 20 males is \(35.25\) minutes, indicating half of the males have exercise times less than or equal to this value, and half have times greater than or equal to it. The interquartile range (IQR) is \(70.0\) minutes, showing that the middle 50% of males have exercise times between \(8.0\) and \(78.0\) minutes, providing an insight into the variability of the exercise durations among these university students.

Step by step solution

01

Arrange the Data in Ascending Order

First, we sort the data in ascending order so it will be easier to find the median, first quartile, and third quartile. The sorted data: \[ 0.0,\, 0.0,\, 0.0,\, 0.0,\, 7.5,\, 8.0,\, 8.0,\, 9.0,\, 28.5,\, 34.5,\, 36.0,\, 43.5,\, 57.0,\, 57.0,\, 78.0,\, 91.5,\, 142.5,\, 151.5,\, 199.5, \]
02

Calculate Median

Now, we need to find the middle value. Since there are 20 data points, the median will be the average of the 10th and 11th value. Median = \(\frac{34.5 + 36.0}{2}\) = \(35.25\)
03

Find Quartiles

We need to find Q1 and Q3, by taking the middle values of the lower half and upper half of the sorted data, respectively. Q1 is the middle value of the lower half (first 10 values): \(Q1 = 8.0\) Q3 is the middle value of the upper half (last 10 values): \(Q3 = 78.0\)
04

Calculate Interquartile Range

Now, we will find the IQR by subtracting Q1 from Q3: IQR = \(Q3 - Q1 = 78.0 - 8.0 = 70.0\)
05

Interpret the Results

The median of the data is \(35.25\) minutes, which means that half of the males have weekday exercise times less than or equal to \(35.25\) minutes, and the other half have weekday exercise times greater than or equal to \(35.25\) minutes. This value represents the central tendency of the exercise time. The interquartile range (IQR) is \(70.0\) minutes, which gives us an idea of the spread of the data. With an IQR of \(70.0\) minutes, we can say that the middle 50% of the males have weekday exercise times ranging from \(8.0\) to \(78.0\) minutes. This value provides an indication of how varied the exercise times are among the university students.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Median Calculation
Calculating the median involves finding the middle value of a data set. The median provides a measure of central tendency, offering a sense of the typical value in the data. First, arrange all the values from smallest to largest. In our data set of 20 values, we order them:
  • 0.0, 0.0, 0.0, 0.0, 7.5, 8.0, 8.0, 9.0, 28.5, 34.5,
  • 36.0, 43.5, 57.0, 57.0, 78.0, 91.5, 142.5, 151.5, 199.5
With 20 data points, calculate the median by averaging the 10th and 11th numbers:
\[\text{Median} = \frac{34.5 + 36.0}{2} = 35.25\]Such a median of 35.25 minutes implies that half the students exercised less than or equal to this time, while the other half exercised more.
Interquartile Range
The interquartile range (IQR) measures the spread of the middle 50% of data points, acting as an indicator of variability. To find the IQR, determine the first (Q1) and third (Q3) quartiles. These are essentially the medians of the first and second halves of your data set:
  • Q1: The median of the first 10 sorted numbers: 0.0 to 34.5; Q1 = 8.0.
  • Q3: The median of the last 10 sorted numbers: 36.0 to 199.5; Q3 = 78.0.
Calculate the IQR as follows:
\[\text{IQR} = Q3 - Q1 = 78.0 - 8.0 = 70.0\]This IQR of 70.0 minutes indicates the range within which the middle 50% of exercise times spread out, from 8.0 to 78.0 minutes.
Data Interpretation
Interpreting data involves understanding what the descriptive statistics tell us about the underlying patterns and distributions.
  • Median of 35.25 minutes: This suggests that on average, a typical student spends about 35 minutes exercising during weekdays. The median is less affected by extremely high or low values and therefore gives a central value that's more representative of the group's behavior as a whole.
  • IQR of 70 minutes: A large IQR shows significant variation among students' exercise times. It reveals that while some students might just exercise for a few minutes, others engage in prolonged physical activities.
With these statistics, educators and health professionals can cultivate better health initiatives, understanding the different levels of engagement and promoting strategies that motivate lesser-active students while supporting more active ones.

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