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The owner of a sporting goods store is making a supply purchase for the coming month. Based on past experience, he has constructed a model which shows that customers choose to buy soccer balls over baseball bats and tennis rackets [tex]58\%[/tex] of the time.

The table below shows the results of three weeks of business with breakdowns for how many customers purchased soccer balls, baseball bats, and tennis rackets.

[tex]
\[
\begin{tabular}{|c|c|c|c|}
\hline
\text{Week} & \text{Soccer Balls} & \text{Baseball Bats} & \text{Tennis Rackets} \\
\hline
1 & 85 & 30 & 31 \\
\hline
2 & 110 & 22 & 23 \\
\hline
3 & 64 & 21 & 23 \\
\hline
\end{tabular}
\]
[/tex]

Classify the results for each category as either consistent or inconsistent with the model.

- Week 1 results
- Week 2 results
- Week 3 results
- Total of all 3 weeks' results

[tex]
\[
\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
\text{Consistent with} \\
\text{Model}
\end{tabular} &
\begin{tabular}{c}
\text{Inconsistent with} \\
\text{Model}
\end{tabular} \\
\hline
& \\
\hline
\end{tabular}
\]
[/tex]

Answer :

To determine if the customers' purchasing behavior aligns with the model, we look at the percentage of customers who purchased soccer balls each week and overall, and compare it to the model's prediction of 58%.

Here is how we analyze the data:

1. Week 1 Results:
- Number of customers who bought soccer balls: 85
- Total customers (soccer balls, baseball bats, tennis rackets): 85 + 30 + 31 = 146
- Calculate the percentage: [tex]\( \frac{85}{146} \approx 0.5822 \)[/tex] or [tex]\( 58.22\% \)[/tex]
- Since 58.22% is higher than the model's prediction of 58%, Week 1 is classified as Consistent with the model.

2. Week 2 Results:
- Number of customers who bought soccer balls: 110
- Total customers: 110 + 22 + 23 = 155
- Calculate the percentage: [tex]\( \frac{110}{155} \approx 0.7097 \)[/tex] or [tex]\( 70.97\% \)[/tex]
- Since 70.97% is higher than the model's prediction of 58%, Week 2 is classified as Consistent with the model.

3. Week 3 Results:
- Number of customers who bought soccer balls: 64
- Total customers: 64 + 21 + 23 = 108
- Calculate the percentage: [tex]\( \frac{64}{108} \approx 0.5926 \)[/tex] or [tex]\( 59.26\% \)[/tex]
- Since 59.26% is higher than the model's prediction of 58%, Week 3 is classified as Consistent with the model.

4. Total of All 3 Weeks' Results:
- Total number of customers who bought soccer balls: 85 + 110 + 64 = 259
- Total customers over the three weeks: 146 + 155 + 108 = 409
- Calculate the overall percentage: [tex]\( \frac{259}{409} \approx 0.6332 \)[/tex] or [tex]\( 63.32\% \)[/tex]
- Since 63.32% is higher than the model's prediction of 58%, the total results for all three weeks are classified as Consistent with the model.

In summary, the results for Week 1, Week 2, Week 3, and the total for all three weeks are all consistent with the model's expectation that 58% of customers will buy soccer balls.

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