The distance (in km) of 40 engineers from their residence to their place of work were found as follows:
5 3 10 20 25 11 13 7 12 31 19 10 12 17 18 11 32 17 16 2 7 9 7 8 3 5 12 15 18 3 12 14 2 9 6 15 15 7 6 12
What is the empirical probability that an engineer lives:
more than or equal to 7 km from her place of work?

Answer

To find the empirical probability that an engineer lives more than or equal to 7 km from her place of work, we can follow these steps:

### Step 1: List the distances
We have the following distances (in km) of 40 engineers from their residence to their place of work:
5, 3, 10, 20, 25, 11, 13, 7, 12, 31, 19, 10, 12, 17, 18, 11, 32, 17, 16, 2, 7, 9, 7, 8, 3, 5, 12, 15, 18, 3, 12, 14, 2, 9, 6, 15, 15, 7, 6, 12

### Step 2: Count the total number of engineers
The total number of engineers is given as 40.

### Step 3: Identify the favorable outcomes
We need to count how many engineers live at a distance of 7 km or more from their place of work. We will go through the list and count the distances that are greater than or equal to 7 km.

- Distances ≥ 7 km:
- 10, 20, 25, 11, 13, 7, 12, 31, 19, 10, 12, 17, 18, 11, 32, 17, 16, 7, 9, 7, 8, 12, 15, 18, 12, 14, 9, 15, 15, 7, 12
- Count of distances ≥ 7 km:
- 10, 20, 25, 11, 13, 7, 12, 31, 19, 10, 12, 17, 18, 11, 32, 17, 16, 7, 9, 7, 8, 12, 15, 18, 12, 14, 9, 15, 15, 7, 12
- Total = 31

### Step 4: Calculate the empirical probability
The empirical probability (P) that an engineer lives more than or equal to 7 km from her place of work is given by the formula:

\[ P(\text{distance} \geq 7) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]

Substituting the values we found:

\[ P(\text{distance} \geq 7) = \frac{31}{40} \]

### Final Answer
The empirical probability that an engineer lives more than or equal to 7 km from her place of work is \( \frac{31}{40} \).

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