Question

a large school system summarizes the number of teachers at 51 schools in the area. frequency table: intervals (20–30, 30–40, 40–50, 50–60, 60–70, 70–80, 80–90, 90–100, 100–110, 110–120) with counts (3, 3, 5, 6, 6, 4, 4, 9, 8, 1) minimum: 20 teachers; q1: 40; median: 65 teachers; q3: 95; maximum: 110. 1. create a box plot that shows this information. axis: 0, 20, 40, 60, 80, 100, 120 (teachers per school). 2. create a histogram that shows this information. axis: 0, 20, 40, 60, 80, 100, 120 (teachers per school); y - axis 0–10. 3. which of these data displays most easily shows how many schools have at least 100 teachers per school? explain your reasoning.

Explanation:

1. Creating the Box Plot

Step 1: Identify the five - number summary

We are given the minimum value \(= 20\), \(Q_1 = 40\), median \(= 65\), \(Q_3=95\), and maximum \(= 110\).

Step 2: Draw the number line

The number line is already provided with a range from 0 to 120, labeled "Teachers per school".

Step 3: Draw the box

• The left end of the box is at \(Q_1 = 40\) and the right end of the box is at \(Q_3=95\).
• Draw a vertical line inside the box at the median value of 65.

Step 4: Draw the whiskers

• The left whisker extends from the minimum value (20) to \(Q_1\) (40).
• The right whisker extends from \(Q_3\) (95) to the maximum value (110).

2. Creating the Histogram

Step 1: Identify the intervals and frequencies

The intervals (classes) and their corresponding frequencies (number of schools) are:

• \(20 - 30\): Frequency \(= 3\)
• \(30 - 40\): Frequency \(= 2\)
• \(40 - 50\): Frequency \(= 5\)
• \(50 - 60\): Frequency \(= 6\)
• \(60 - 70\): Frequency \(= 6\)
• \(70 - 80\): Frequency \(= 4\)
• \(80 - 90\): Frequency \(= 4\)
• \(90 - 100\): Frequency \(= 7\)
• \(100 - 110\): Frequency \(= 8\)
• \(110 - 120\): Frequency \(= 1\)

Step 2: Draw the axes

The x - axis is labeled "Teachers per school" with intervals from 0 - 120 (with the class intervals as above). The y - axis is labeled "Frequency" (number of schools) with a range from 0 to 10.

Step 3: Draw the bars

For each interval, draw a bar whose height is equal to the frequency of that interval. For example:

• For the interval \(20 - 30\), the bar height is 3.
• For the interval \(30 - 40\), the bar height is 2.
• For the interval \(40 - 50\), the bar height is 5.
• And so on for the remaining intervals.

3. Determining the Best Data Display

Step 1: Analyze the box - plot

A box - plot is mainly used to show the spread of data (range, inter - quartile range) and the median. It does not show the frequency of data in specific intervals (like the number of schools with at least 100 teachers) directly.

Step 2: Analyze the histogram

A histogram is a bar - graph that shows the frequency (number of data points) in each interval (class). To find the number of schools with at least 100 teachers, we can look at the intervals \(100 - 110\) and \(110 - 120\) in the histogram. The frequency for \(100 - 110\) is 8 and for \(110 - 120\) is 1. We can sum these frequencies (\(8 + 1=9\)) to find the number of schools with at least 100 teachers. Since the histogram is designed to display the frequency of data in different intervals, it is the best display to show how many schools have at least 100 teachers per school.

Answer:

s:

1. The box - plot is drawn with the box from 40 to 95, a vertical line at 65, left whisker from 20 to 40, and right whisker from 95 to 110 on the given number line.
2. The histogram is drawn with bars of heights corresponding to the frequencies of each interval (as calculated above) on the given axes.
3. The histogram most easily shows how many schools have at least 100 teachers per school. This is because a histogram displays the frequency (number of schools) in each interval, so we can directly look at the intervals \(100 - 110\) and \(110 - 120\) and sum their frequencies to find the number of schools with at least 100 teachers. A box - plot is more focused on showing the spread and median of the data rather than the frequency in specific intervals.