Triangle Midsegment Theorem worksheet with construction and calculation exercises.

Problem Overview:

The task involves analyzing the ages of 20 math teachers at a Virginia high school. The provided data is as follows:
```
54, 58, 60, 43, 35, 32, 47, 28, 33, 42, 65, 73, 44, 27, 32, 35, 28, 25, 48, 33
```

We are tasked with:
1. Constructing a dotplot for the data.
2. Constructing a stemplot for the data.
3. Completing a frequency table and constructing a histogram.
4. Describing the shape of the distribution.
5. Describing the spread of the distribution.
6. Identifying the center of the distribution.
7. Checking for any outliers.
8. Determining the width of each class in the histogram.
9. Deciding whether the data can be represented by a pie graph and explaining why or why not.

---

Solution:

#### 1. Construct a Dotplot for the Data
A dotplot is a simple graphical display where each data point is represented by a dot above a number line.

- Arrange the data in ascending order:
`25, 27, 28, 28, 32, 32, 33, 33, 35, 35, 42, 43, 44, 47, 48, 54, 58, 60, 65, 73`
- Plot each value on a number line with dots.

Dotplot:
```
20 |
25 | *
26 |
27 | *
28 | * *
29 |
30 |
31 |
32 | * *
33 | * *
34 |
35 | * *
36 |
37 |
38 |
39 |
40 |
41 |
42 | *
43 | *
44 | *
45 |
46 |
47 | *
48 | *
49 |
50 |
51 |
52 |
53 |
54 | *
55 |
56 |
57 |
58 | *
59 |
60 | *
61 |
62 |
63 |
64 |
65 | *
66 |
67 |
68 |
69 |
70 |
71 |
72 |
73 | *
```

#### 2. Construct a Stemplot for the Data
A stemplot (also called a stem-and-leaf plot) separates each number into a "stem" (the leading digit(s)) and a "leaf" (the trailing digit).

- Stems: Tens place (`2, 3, 4, 5, 6, 7`)
- Leaves: Ones place

Stemplot:
```
2 | 5 7 8 8
3 | 2 2 3 3 5 5
4 | 2 3 4 7 8
5 | 4 8
6 | 0 5
7 | 3
```

#### 3. Complete the Frequency Table and Construct the Corresponding Histogram
First, we define class intervals for the frequency table. A reasonable choice for class width is 10, given the range of the data (from 25 to 73).

Class Intervals:
- 25 to < 35
- 35 to < 45
- 45 to < 55
- 55 to < 65
- 65 to < 75

Frequency Table:
| Class | Count |
|-------------|-------|
| 25 to < 35 | 7 |
| 35 to < 45 | 6 |
| 45 to < 55 | 4 |
| 55 to < 65 | 3 |
| 65 to < 75 | 2 |

Histogram:
- Draw a bar for each class interval with height proportional to the count.
- X-axis: Class intervals
- Y-axis: Frequency

#### 4. Describe the Shape of the Distribution
From the dotplot and histogram:
- The distribution is roughly symmetric.
- There is no strong skewness to the left or right.
- The data is relatively balanced around the center.

Shape: Roughly symmetric.

#### 5. Describe the Spread of the Distribution
The spread can be described using the range and interquartile range (IQR):
- Range: Maximum value - Minimum value = 73 - 25 = 48
- Quartiles:
- Q1 (25th percentile): Around 32
- Q3 (75th percentile): Around 54
- IQR = Q3 - Q1 = 54 - 32 = 22

The data spans a range of 48 years, with the middle 50% of the data falling within a 22-year range.

Spread: The data has a moderate spread, with a range of 48 and an IQR of 22.

#### 6. What is the Center of the Distribution?
The center can be identified using the median or mean:
- Median: The middle value when the data is ordered. For 20 data points, the median is the average of the 10th and 11th values:
`(35 + 42) / 2 = 38.5`
- Mean: Sum of all values divided by the number of values:
\[
\text{Mean} = \frac{25 + 27 + 28 + 28 + 32 + 32 + 33 + 33 + 35 + 35 + 42 + 43 + 44 + 47 + 48 + 54 + 58 + 60 + 65 + 73}{20} = 42.1
\]

Center: The median is 38.5, and the mean is approximately 42.1.

#### 7. Are There Any Obvious Outliers?
Outliers are typically defined as values that fall outside the range:
\[
Q1 - 1.5 \times \text{IQR} \quad \text{or} \quad Q3 + 1.5 \times \text{IQR}
\]
- Q1 = 32, Q3 = 54, IQR = 22
- Lower boundary: \(32 - 1.5 \times 22 = 32 - 33 = -1\) (no values below this)
- Upper boundary: \(54 + 1.5 \times 22 = 54 + 33 = 87\) (no values above this)

There are no values outside these boundaries, so there are no obvious outliers.

Outliers: No outliers.

#### 8. What is the Width of Each Class in the Histogram?
The class intervals used in the histogram are:
- 25 to < 35
- 35 to < 45
- 45 to < 55
- 55 to < 65
- 65 to < 75

Each class has a width of 10.

Width: 10

#### 9. Could This Data Set Be Represented by a Pie Graph? Why or Why Not?
A pie graph is used to show proportions of categorical data. Since the data here represents numerical ages (continuous data), a pie graph would not be appropriate. Pie graphs are better suited for displaying parts of a whole in categorical variables.

Answer: No, because the data is numerical and continuous, not categorical.

---

Final Answers:

1. Dotplot: See constructed dotplot above.
2. Stemplot: See constructed stemplot above.
3. Frequency Table and Histogram: See constructed table and description above.
4. Shape: Roughly symmetric.
5. Spread: Range = 48, IQR = 22.
6. Center: Median = 38.5, Mean ≈ 42.1.
7. Outliers: No outliers.
8. Width of Each Class: 10.
9. Pie Graph: No, because the data is numerical and continuous.

Final Boxed Answer:
\[
\boxed{\text{Roughly symmetric, moderate spread, center around 38.5, no outliers, class width 10, not suitable for pie graph}}
\]