Question

box plot worksheet. reading times a teacher recorded the amount of time (in minutes) 100 students spent reading each night. the data set has already been summarized into a five - number summary: minimum: 5 q1: 15 median: 25 q3: 35 maximum: 70 tasks: 1. construct a box plot using the five - number summary. 2. interpret what each of the five values tells you about students’ reading habits. fill in the table below with your interpretations.

value	interpretation in the situation
q1 (15)
median (25)
q3 (35)
maximum (70)

guiding questions: - what does the median say about a “typical” student? - what does q1 tell us about the lower group of students? - what can you say about the top readers (q3 and maximum)? - does the maximum look like an outlier? why or why not?

Answer:

Task 2: Interpreting the Five - Number Summary

Minimum (5)

• Explanation: The minimum value in a data set represents the smallest observation. In the context of students' reading time, it tells us the least amount of time any student spent reading each night.
• Interpretation: At least one student spent 5 minutes reading each night, which is the shortest reading time among all 100 students.

Q1 (15)

• Explanation: The first quartile (Q1) is the value below which 25% of the data lies. So, 25% of the students have a reading time less than or equal to Q1.
• Interpretation: 25% of the 100 students (i.e., 25 students) spent 15 minutes or less reading each night. This gives us an idea about the reading habits of the lower - performing (in terms of reading time) quarter of the students.

Median (25)

• Explanation: The median is the middle value of a sorted data set. It divides the data into two equal halves, with 50% of the data below it and 50% above it. For a "typical" student, the median gives a central tendency measure that is not affected by extreme values.
• Interpretation: 50% of the students spent 25 minutes or less reading each night, and 50% spent 25 minutes or more. So, a "typical" student (in the sense of the middle - performing student) spent around 25 minutes reading each night.

Q3 (35)

• Explanation: The third quartile (Q3) is the value below which 75% of the data lies. So, 75% of the students have a reading time less than or equal to Q3, and 25% have a reading time greater than Q3.
• Interpretation: 75% of the 100 students (i.e., 75 students) spent 35 minutes or less reading each night. This helps us understand the reading habits of the upper - middle and lower - performing students combined, and also gives an idea about the top 25% of the students (who read more than 35 minutes).

Maximum (70)

• Explanation: The maximum value in a data set represents the largest observation. In the context of students' reading time, it tells us the most amount of time any student spent reading each night.
• Interpretation: At least one student spent 70 minutes reading each night, which is the longest reading time among all 100 students. To check if it's an outlier, we can use the inter - quartile range (IQR) method. The IQR is calculated as \(IQR = Q3 - Q1\). Here, \(IQR=35 - 15 = 20\). The lower bound for non - outliers is \(Q1-1.5\times IQR=15 - 1.5\times20=15 - 30=- 15\) (since time can't be negative, we consider 0 as the lower practical bound), and the upper bound for non - outliers is \(Q3 + 1.5\times IQR=35+1.5\times20=35 + 30 = 65\). Since 70>65, the maximum value of 70 minutes is an outlier. It means that there is at least one student whose reading time is much higher than the majority of the students.

Filling the Table

Value	Interpretation in the Situation
Q1 (15)	25% of students read 15 minutes or less each night.
Median (25)	50% of students read 25 minutes or less, 50% read 25 minutes or more (a "typical" student reads ~25 minutes).
Q3 (35)	75% of students read 35 minutes or less each night.
Maximum (70)	The longest reading time among students is 70 minutes (it is an outlier as 70 > 65, where 65 is \(Q3 + 1.5\times IQR\)).

Task 1: Constructing the Box Plot

Step 1: Draw a Number Line

• Draw a horizontal number line that covers the range from the minimum value (5) to the maximum value (70). Label the relevant values: 5, 15, 25, 35, 70.

Step 2: Draw the Box

• The box in a box - plot spans from Q1 to Q3. So, draw a rectangle (the box) with the left - hand side at Q1 = 15 and the right - hand side at Q3 = 35.
• Draw a vertical line inside the box at the median value (25) to represent the median.

Step 3: Draw the Whiskers

• The lower whisker extends from the minimum value (5) to Q1 (15). Draw a line from 5 to 15, and add a dot or a short line at 5 to represent the minimum.
• The upper whisker: First, we need to check for outliers. As calculated before, the upper bound for non - outliers is 65. Since the maximum value (70) is an outlier, the upper whisker extends from Q3 (35) to the largest non - outlier value (if there was one less than 65). But since we only know the five - number summary, we can draw the upper whisker from Q3 (35) to 65 (the upper non - outlier bound) and then represent the outlier (70) as a separate dot above 65 on the number line.

Final Answers (for Task 2 Table)

Value	Interpretation in the Situation
Q1 (15)	25% of students read ≤15 minutes.
Median (25)	50% read ≤25, 50% ≥25 (typical ~25 min).
Q3 (35)	75% of students read ≤35 minutes.
Maximum (70)	Longest reading time is 70 minutes (outlier, as 70 > 65).