To determine whether not regularly hiking and living near a lake are independent events, we need to compare the probability of not regularly hiking given that a person lives near a lake to the probability of not regularly hiking.
From the table:
- The total number of people surveyed is 200.
- The total number of people who do not hike regularly is 100.
- The total number of people living near a lake is 48.
- The number of people who do not hike regularly and live near a lake is 24.
1. Calculate the probability of not regularly hiking (P(Not Hike)):
[tex]\[ P(\text{Not Hike}) = \frac{\text{Total does not hike}}{\text{Total population}} = \frac{100}{200} = 0.5 \][/tex]
2. Calculate the probability of living near a lake (P(Near Lake)):
[tex]\[ P(\text{Near Lake}) = \frac{\text{Total near lake}}{\text{Total population}} = \frac{48}{200} = 0.24 \][/tex]
3. Calculate the probability of not regularly hiking given that a person lives near a lake (P(Not Hike | Near Lake)):
[tex]\[ P(\text{Not Hike | Near Lake}) = \frac{\text{Does not hike near lake}}{\text{Total near lake}} = \frac{24}{48} = 0.5 \][/tex]
4. Comparison to determine independence:
[tex]\[ P(\text{Not Hike}) = P(\text{Not Hike | Near Lake}) \][/tex]
Since both probabilities are equal (0.5), not regularly hiking and living near a lake are independent events.
Thus, the complete statement is:
Not regularly hiking and living near a lake are independent events because the probability of not regularly hiking given that a person lives near a lake is equal to the probability of not regularly hiking.
