To determine if there is an association between the number of miles an individual runs per week and how many pairs of running shoes they purchase per year, we can create a conditional relative frequency table based on the given data.
First, let’s summarize the data provided:
-
Runs more than 5 miles per week:
- Purchases more than 2 pairs per year: 42
- Purchases less than 2 pairs per year: 28
-
Runs less than 6 miles per week:
- Purchases more than 2 pairs per year: 15
- Purchases less than 2 pairs per year: 32
Now, let's calculate the total for each category:
- Runs more than 5 miles per week: \( 42 + 28 = 70 \)
- Runs less than 6 miles per week: \( 15 + 32 = 47 \)
Now we can create the conditional relative frequency table, giving us the proportion of buyers versus non-buyers within each running distance category:
| Running Distance | Purchases > 2 pairs per year | Purchases < 2 pairs per year | Total | |-----------------------------|---------------------------|---------------------------|-------| | Runs more than 5 miles | \( \frac{42}{70} = 0.60 \) | \( \frac{28}{70} = 0.40 \) | 70 | | Runs less than 6 miles | \( \frac{15}{47} \approx 0.32 \) | \( \frac{32}{47} \approx 0.68 \) | 47 |
Now summarizing the percentages:
- Runs more than 5 miles per week: 60% purchase more than 2 pairs; 40% purchase less than 2 pairs.
- Runs less than 6 miles per week: About 32% purchase more than 2 pairs; about 68% purchase less than 2 pairs.
Reasoning: The data shows a clear association between the number of miles run per week and the number of shoes purchased per year. Runners who run more than 5 miles per week are more likely to purchase more than 2 pairs of shoes per year (60%) compared to those who run less than 6 miles (32%). The higher percentage of shoe purchases among the more active runners suggests that their running habits influence their purchasing behavior, thus indicating a positive association between the two factors.