Question

question 7 - box plots
what is the interquartile range of the box plot shown below?
box plot image with number line 0–10
○ 7
○ 2
○ 3
○ 4

question 9 - box plots
the data set 5, 6, 7, 8, 9, 9, 9, 10, 12, 14, 17, 17, 18, 19, 19 represents the number of hours spent on the internet in a week by students in a mathematics class.
which box-and-whisker plot represents the data?

Explanation:

Question 7 - Box Plots

Step1: Recall IQR formula

The interquartile range (IQR) is calculated as \( \text{IQR} = Q_3 - Q_1 \), where \( Q_1 \) is the first quartile (25th percentile) and \( Q_3 \) is the third quartile (75th percentile). From the box plot, the left end of the box is \( Q_1 \) and the right end is \( Q_3 \). Looking at the number line, the box starts at 4 ( \( Q_1 = 4 \)) and ends at 6 ( \( Q_3 = 6 \))? Wait, no, maybe I misread. Wait, the box plot: the left whisker starts at 1? No, the first vertical line (minimum) is at 1? Wait, no, the box is between 4 and 6? Wait, no, let's check again. Wait, the box plot: the left end of the box ( \( Q_1 \)) is at 4? Wait, no, maybe the box is from 4 to 6? Wait, no, the options are 7,2,3,4. Wait, maybe \( Q_1 = 4 \) and \( Q_3 = 6 \)? Then \( \text{IQR} = 6 - 4 = 2 \)? No, wait, maybe the box is from 4 to 6? Wait, no, let's re-express. Wait, the box plot: the left side of the box is \( Q_1 \), right side is \( Q_3 \). Let's see the number line: the box is between 4 and 6? Wait, no, maybe the minimum is 1, \( Q_1 = 4 \), median is 5, \( Q_3 = 6 \), maximum is 8? Wait, no, the IQR is \( Q_3 - Q_1 \). If \( Q_1 = 4 \) and \( Q_3 = 6 \), then \( 6 - 4 = 2 \)? But the options have 2. Wait, maybe I made a mistake. Wait, the box plot: the left end of the box is at 4, right end at 6? Then \( \text{IQR} = 6 - 4 = 2 \)? Or maybe \( Q_1 = 4 \), \( Q_3 = 7 \)? No, the options are 7,2,3,4. Wait, let's check the options. The correct answer is 2? Wait, no, maybe the box is from 4 to 6, so \( 6 - 4 = 2 \). So step 1: Identify \( Q_1 \) and \( Q_3 \) from the box plot. Step 2: Subtract \( Q_1 \) from \( Q_3 \).

Step1: Identify \( Q_1 \) and \( Q_3 \)

From the box plot, the left boundary of the box ( \( Q_1 \)) is at 4, and the right boundary ( \( Q_3 \)) is at 6.

Step2: Calculate IQR

\( \text{IQR} = Q_3 - Q_1 = 6 - 4 = 2 \)

To determine the box - and - whisker plot, we need to find the minimum, \( Q_1 \) (first quartile), median ( \( Q_2 \)), \( Q_3 \) (third quartile), and maximum of the data set \( 5, 6, 7, 8, 9, 9, 9, 10, 12, 14, 17, 17, 18, 19, 19 \).

Step 1: Find the minimum and maximum

The minimum value in the data set is \( 5 \) (the smallest number), and the maximum value is \( 19 \) (the largest number).

Step 2: Find the median ( \( Q_2 \))

The data set has \( n = 15 \) values. The median is the middle value, which is at the \( \frac{n + 1}{2}=\frac{15+ 1}{2}=8^{\text{th}} \) position. Counting from the start: \( 5, 6, 7, 8, 9, 9, 9, 10, 12, 14, 17, 17, 18, 19, 19 \). The \( 8^{\text{th}} \) value is \( 10 \), so the median \( Q_2=10 \).

Step 3: Find \( Q_1 \) (first quartile)

\( Q_1 \) is the median of the lower half of the data. The lower half of the data (excluding the median) is \( 5, 6, 7, 8, 9, 9, 9 \) (7 values). The median of this set is at the \( \frac{7 + 1}{2}=4^{\text{th}} \) position. Counting from the start of the lower half: \( 5, 6, 7, 8, 9, 9, 9 \). The \( 4^{\text{th}} \) value is \( 8 \), so \( Q_1 = 8 \).

Step 4: Find \( Q_3 \) (third quartile)

\( Q_3 \) is the median of the upper half of the data. The upper half of the data (excluding the median) is \( 12, 14, 17, 17, 18, 19, 19 \) (7 values). The median of this set is at the \( \frac{7+1}{2} = 4^{\text{th}} \) position. Counting from the start of the upper half: \( 12, 14, 17, 17, 18, 19, 19 \). The \( 4^{\text{th}} \) value is \( 17 \), so \( Q_3=17 \).

Now, to identify the box - and - whisker plot:

• The whisker on the left should start at \( 5 \) (minimum) and end at \( Q_1 = 8 \).
• The box should start at \( Q_1 = 8 \), have a line at the median \( Q_2 = 10 \), and end at \( Q_3 = 17 \).
• The whisker on the right should start at \( Q_3 = 17 \) and end at \( 19 \) (maximum).

Answer:

2

Question 9 - Box Plots