Use the table to answer the question.

Question

Purchases More than 2 Pairs per Year

Purchases 2 Pairs or Less per Year

Total
Runs More than 5 Miles per Week

42 28 70
Runs Less than 5 Miles per Week

15 32 47
A running shoe company is collecting data on long-distance runners. The company’s research analyst asked runners how many pairs of running shoes they purchase each year and how many miles they run each week, on average. The results are summarized in the table. 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. Create a conditional relative frequency table of the data. Use your table to support your reasoning. Explain your reasoning in 3–5 sentences.

(4 points)

Answer 1

To analyze the association between the number of miles run per week and the number of pairs of running shoes purchased per year, we first create a conditional relative frequency table by calculating the relative frequencies based on the rows (running miles) and the columns (shoe purchases).

Step 1: Calculate the totals.

• For runners who run more than 5 miles per week: 70 total
• For runners who run less than 5 miles per week: 47 total

Step 2: Calculate the relative frequencies (by row).

For runners who run more than 5 miles per week:

• Relative frequency of purchasing more than 2 pairs: \( \frac{42}{70} = 0.60 \)
• Relative frequency of purchasing 2 pairs or less: \( \frac{28}{70} = 0.40 \)

For runners who run less than 5 miles per week:

• Relative frequency of purchasing more than 2 pairs: \( \frac{15}{47} \approx 0.32 \)
• Relative frequency of purchasing 2 pairs or less: \( \frac{32}{47} \approx 0.68 \)

Conditional Relative Frequency Table:

| Running Miles per Week | Purchases More than 2 Pairs | Purchases 2 Pairs or Less | Total | |----------------------------------|------------------------------|---------------------------|-------| | More than 5 | 0.60 | 0.40 | 1.00 | | Less than 5 | 0.32 | 0.68 | 1.00 |

Step 3: Analyze the data.

From the relative frequencies, we observe that a higher proportion (60%) of runners who run more than 5 miles per week purchase more than 2 pairs of shoes, compared to only 32% of those who run less than 5 miles per week. This suggests a notable association: runners who log more miles are likely to invest in more pairs of running shoes than those who run fewer miles.

Thus, we can conclude there is an association between the number of miles run per week and the number of pairs of shoes purchased per year, as the data indicates that more dedicated runners tend to buy more athletic footwear.