To determine which statement is true regarding the remaining vacation days for Matt and Linda, we will calculate the mean and median values for both of them.
### Step-by-Step Calculation:
#### 1. Data for Matt and Linda:
- Matt's vacation days at the end of each quarter: [13, 11, 7, 4]
- Linda's vacation days at the end of each quarter: [12, 11, 7, 0]
#### 2. Calculate the Mean:
- Mean for Matt:
[tex]\[
\text{mean}_{\text{Matt}} = \frac{13 + 11 + 7 + 4}{4}
\][/tex]
Simplifying the numerator:
[tex]\[
13 + 11 = 24
\][/tex]
[tex]\[
24 + 7 = 31
\][/tex]
[tex]\[
31 + 4 = 35
\][/tex]
Therefore,
[tex]\[
\text{mean}_{\text{Matt}} = \frac{35}{4} = 8.75
\][/tex]
- Mean for Linda:
[tex]\[
\text{mean}_{\text{Linda}} = \frac{12 + 11 + 7 + 0}{4}
\][/tex]
Simplifying the numerator:
[tex]\[
12 + 11 = 23
\][/tex]
[tex]\[
23 + 7 = 30
\][/tex]
[tex]\[
30 + 0 = 30
\][/tex]
Therefore,
[tex]\[
\text{mean}_{\text{Linda}} = \frac{30}{4} = 7.5
\][/tex]
#### 3. Calculate the Median:
- Median for Matt:
The ordered list of Matt’s vacation days is [4, 7, 11, 13]. The median is the average of the two middle numbers:
[tex]\[
\text{median}_{\text{Matt}} = \frac{7 + 11}{2} = \frac{18}{2} = 9.0
\][/tex]
- Median for Linda:
The ordered list of Linda’s vacation days is [0, 7, 11, 12]. The median is the average of the two middle numbers:
[tex]\[
\text{median}_{\text{Linda}} = \frac{7 + 11}{2} = \frac{18}{2} = 9.0
\][/tex]
### Comparison and Conclusion:
- Mean Comparison:
[tex]\[
\text{mean}_{\text{Matt}} = 8.75 > 7.5 = \text{mean}_{\text{Linda}}
\][/tex]
- Median Comparison:
[tex]\[
\text{median}_{\text{Matt}} = 9.0 = 9.0 = \text{median}_{\text{Linda}}
\][/tex]
Given these calculations, we can confirm that:
- The mean number of vacation days Matt has remaining is greater than the mean number of vacation days Linda has remaining.
- The median number of vacation days remaining for both Matt and Linda is the same.
Thus, the correct statement is:
D. The mean number of vacation days Matt has remaining is greater than the mean number of vacation days Linda has remaining.
