Question

The heights, in feet, of the trees for sale at two nurseries are shown below.

Yard Works: 7, 9, 7, 12, 5
The Grow Station: 9, 11, 6, 12, 7

Which statements are true regarding the measures of center and variability of these data sets? Select three choices.
The mean of the tree heights at Yard Works is less than the mean of the tree heights at The Grow Station.
The median of the tree heights at Yard Works is greater than the median of the tree heights at The Grow Station.
The range of the tree heights at Yard Works is greater than the range of the tree heights at The Grow Station.
The mean absolute deviation of the tree heights at Yard Works is greater than the mean absolute deviation of the tree heights at The Grow Station.
The mean absolute deviation of the tree heights at Yard Works is equal to the mean absolute deviation of the tree heights at The Grow Station.

Answer 1

To determine which statements are true, we will calculate the necessary statistics for both data sets.

**Yard Works: 7, 9, 7, 12, 5**
1. **Mean**:
\[
\text{Mean} = \frac{(7 + 9 + 7 + 12 + 5)}{5} = \frac{40}{5} = 8
\]

2. **Median**:
- Ordered values: 5, 7, 7, 9, 12
- Median (middle value): 7

3. **Range**:
\[
\text{Range} = 12 - 5 = 7
\]

4. **Mean Absolute Deviation (MAD)**:
\[
\text{MAD} = \frac{( |7-8| + |9-8| + |7-8| + |12-8| + |5-8| )}{5} = \frac{(1 + 1 + 1 + 4 + 3)}{5} = \frac{10}{5} = 2
\]

---

**The Grow Station: 9, 11, 6, 12, 7**
1. **Mean**:
\[
\text{Mean} = \frac{(9 + 11 + 6 + 12 + 7)}{5} = \frac{45}{5} = 9
\]

2. **Median**:
- Ordered values: 6, 7, 9, 11, 12
- Median (middle value): 9

3. **Range**:
\[
\text{Range} = 12 - 6 = 6
\]

4. **Mean Absolute Deviation (MAD)**:
\[
\text{MAD} = \frac{( |9-9| + |11-9| + |6-9| + |12-9| + |7-9| )}{5} = \frac{(0 + 2 + 3 + 3 + 2)}{5} = \frac{10}{5} = 2
\]

Let's summarize the findings:

- **Yard Works**: Mean = 8, Median = 7, Range = 7, MAD = 2
- **The Grow Station**: Mean = 9, Median = 9, Range = 6, MAD = 2

Now evaluate each statement:

1. **The mean of the tree heights at Yard Works is less than the mean of the tree heights at The Grow Station.**
- This is **True** (8 < 9).

2. **The median of the tree heights at Yard Works is greater than the median of the tree heights at The Grow Station.**
- This is **False** (7 < 9).

3. **The range of the tree heights at Yard Works is greater than the range of the tree heights at The Grow Station.**
- This is **True** (7 > 6).

4. **The mean absolute deviation of the tree heights at Yard Works is greater than the mean absolute deviation of the tree heights at The Grow Station.**
- This is **False** (2 = 2).

5. **The mean absolute deviation of the tree heights at Yard Works is equal to the mean absolute deviation of the tree heights at The Grow Station.**
- This is **True** (2 = 2).

Thus, the three true statements are:

1. The mean of the tree heights at Yard Works is less than the mean of the tree heights at The Grow Station.
3. The range of the tree heights at Yard Works is greater than the range of the tree heights at The Grow Station.
5. The mean absolute deviation of the tree heights at Yard Works is equal to the mean absolute deviation of the tree heights at The Grow Station.