Let's solve the problem step by step.
First, let's calculate the sample mean for the number of years worked to the nearest integer.
We start by listing the years and their corresponding number of employees:
- 10 years: 26 employees
- 11 years: 29 employees
- 12 years: 3 employees
- 13 years: 10 employees
- 14 years: 3 employees
- 15 years: 17 employees
- 16 years: 13 employees
- 17 years: 15 employees
- 18 years: 15 employees
- 19 years: 23 employees
- 20 years: 13 employees
Step 1: List the years.
[tex]\[ \text{Years} = [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] \][/tex]
Step 2: Calculate the sample mean of the number of years worked.
Sample mean is given by the sum of the years divided by the number of different year categories.
[tex]\[ \text{Sample Mean} = \frac{\sum (\text{Years})}{\text{Number of Years}} \][/tex]
The sum of the years is:
[tex]\[ 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 175 \][/tex]
The number of different year categories is 11 (from 10 to 20).
So we calculate the mean:
[tex]\[ \text{Sample Mean} = \frac{175}{11} \approx 15.909 \][/tex]
Rounding to the nearest integer:
[tex]\[ \text{Rounded Sample Mean} = 16 \][/tex]
Step 3: Calculate the percentage of employees who worked for the company for at least 10 years.
Since all employees in the sample worked for at least 10 years, the calculation is straightforward.
Step 4: Count the total number of employees:
[tex]\[ 26 + 29 + 3 + 10 + 3 + 17 + 13 + 15 + 15 + 23 + 13 = 167 \][/tex]
Step 5: Calculate the percentage of employees who worked for at least 10 years.
[tex]\[ \text{Percentage} = \left( \frac{\text{Total Employees (at least 10 years)}}{\text{Total Employees in Sample}} \right) \times 100 \][/tex]
Since all the employees listed worked for at least 10 years:
[tex]\[ \text{Percentage} = \left( \frac{167}{167} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage} = 100 \][/tex]
Rounding to the nearest integer isn't required as 100 is already an integer.
So, the answers are:
- The sample mean for the number of years worked, rounded to the nearest integer, is [tex]\( \boxed{16} \)[/tex].
- The percentage of the employees in the sample who worked for the company for at least 10 years, rounded to the nearest integer, is [tex]\( \boxed{100} \% \)[/tex].
