Questions: A survey was given to 18 students. One question asked them to rate their college on a scale from 1-5, with 5 being the highest. The results, in miles, are shown in the following table. Use the results to find the first, second, and third quartiles for the data.

A survey was given to 18 students. One question asked them to rate their college on a scale from 1-5, with 5 being the highest. The results, in miles, are shown in the following table. Use the results to find the first, second, and third quartiles for the data.

Transcript text: A survey was given to 18 students. One question asked them to rate their college on a scale from 1-5, with 5 being the highest. The results, in miles, are shown in the following table. Use the results to find the first, second, and third quartiles for the data.

Solution

Solution Steps

Step 1: Sort the Data

The given data representing the distance traveled to attend college is: \[ [32, 10, 66, 70, 32, 80, 26, 18, 42] \]

First, we sort the data: \[ \text{Sorted data} = [10, 18, 26, 32, 32, 42, 66, 70, 80] \]

Step 2: Calculate the First Quartile (Q1)

To find the first quartile \( Q_1 \), we use the formula: \[ \text{Rank} = Q \times (N + 1) = 0.25 \times (9 + 1) = 2.5 \] Since the rank is not an integer, we take the average of the values at positions 2 and 3 in the sorted data: \[ Q_1 = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{18 + 26}{2} = 22.0 \]

Step 3: Calculate the Second Quartile (Q2)

For the second quartile \( Q_2 \): \[ \text{Rank} = Q \times (N + 1) = 0.5 \times (9 + 1) = 5.0 \] The quantile is at position 5, which corresponds to the value: \[ Q_2 = 32 \]

Step 4: Calculate the Third Quartile (Q3)

To find the third quartile \( Q_3 \): \[ \text{Rank} = Q \times (N + 1) = 0.75 \times (9 + 1) = 7.5 \] Again, since the rank is not an integer, we take the average of the values at positions 7 and 8: \[ Q_3 = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{66 + 70}{2} = 68.0 \]