Let's go step by step to answer each part of the question:

(a) Finding the Values of the Three Quartiles

To find the quartiles, we first need to organize the data in ascending order:

**Sorted dataLet's go step by step to answer each part of the question:

(a) Finding the Values of the Three Quartiles

To find the quartiles, we first need to organize the data in ascending order:

**Sorted ** 7, 14, 17, 18, 24, 29, 37, 39, 42, 47, 55, 63

The three quartiles are:

• Q1 (1st quartile): It separates the lower 25% of the data.
• Q2 (2nd quartile or median): It separates the lower 50% of the data.
• Q3 (3rd quartile): It separates the lower 75% of the data.

For 12 data points:

• Q1 is the value that lies at the position $\frac{n+1}{4} = \frac{12+1}{4} = 3.25$, which means we interpolate between the 3rd and 4th values.
• Q2 (Median) is the average of the 6th and 7th values.
• Q3 is the value at the position $\frac{3(n+1)}{4} = \frac{3(12+1)}{4} = 9.75$, which means we interpolate between the 9th and 10th values.

Let's calculate:

1. Q1 (First Quartile):

   • 3rd value = 17
   • 4th value = 18
   • Q1 = $17 + 0.25(18 - 17) = 17.25$

2. Q2 (Median or Second Quartile):

   • 6th value = 29
   • 7th value = 37
   • Q2 = $\frac{29 + 37}{2} = 33$

3. Q3 (Third Quartile):

   • 9th value = 42
   • 10th value = 47
   • Q3 = $42 + 0.75(47 - 42) = 45.75$

(b) Where does the commuting time of 47 fall in relation to the three quartiles?

The quartiles calculated are:

• Q1 = 17.25
• Q2 = 33
• Q3 = 45.75

The commuting time of 47 minutes lies between Q3 (45.75) and the maximum value of 63, which means it falls above the third quartile (Q3).

(c) Find the Interquartile Range (IQR)

The Interquartile Range (IQR) is the difference between the third quartile and the first quartile: $\text{IQR} = Q3 - Q1 = 45.75 - 17.25 = 28.5$

Summary:

• Q1 = 17.25
• Q2 = 33
• Q3 = 45.75
• The commuting time of 47 minutes falls above Q3.
• IQR = 28.5

Would you like more details on any of the calculations, or do you have any questions?

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Here are 5 questions to expand on this:

1. How would the quartile values change if an additional student with a commuting time of 70 minutes was added?
2. What is the effect of outliers on the interquartile range?
3. How can the IQR be used to identify potential outliers in this data set?
4. What is the relationship between quartiles and percentiles?
5. How would you calculate the standard deviation of this data set?

Tip: The interquartile range (IQR) is a measure of statistical dispersion, and it's often used to detect outliers by finding values that lie beyond 1.5 times the IQR above Q3 or below Q1.