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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 that shows 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
- 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 whether the results for each category are consistent or inconsistent with the model, we'll follow these steps.

### Week 1 Results:

1. Total Number of Purchases in Week 1:
- Soccer Balls: 85
- Baseball Bats: 30
- Tennis Rackets: 31

Calculate the total number of items sold in Week 1:
[tex]\[
\text{Total} = 85 + 30 + 31 = 146
\][/tex]

2. Actual Percentage of Soccer Ball Purchases in Week 1:
- Calculate the percentage of customers who bought soccer balls:
[tex]\[
\text{Actual Percentage} = \frac{85}{146} \approx 0.582
\][/tex]

3. Model Prediction:
- The model predicts that [tex]\(58\%\)[/tex] of customers choose to buy soccer balls.

4. Comparison for Consistency:
- Check if the actual percentage is within [tex]\(5\%\)[/tex] of the predicted [tex]\(58\%\)[/tex] (between [tex]\(53\%\)[/tex] and [tex]\(63\%\)[/tex]).
- Since [tex]\(58.2\%\)[/tex] is within this range, the results for Week 1 are consistent with the model.

### Total of All 3 Weeks' Results:

1. Total Purchases Over All Weeks:
- Week 1: 146
- Week 2: Soccer Balls: 110, Baseball Bats: 22, Tennis Rackets: 23
- Week 3: Soccer Balls: 64, Baseball Bats: 21, Tennis Rackets: 23

Calculate the totals for weeks 2 and 3:
[tex]\[
\text{Total Week 2} = 110 + 22 + 23 = 155
\][/tex]
[tex]\[
\text{Total Week 3} = 64 + 21 + 23 = 108
\][/tex]

Calculate the overall total sales:
[tex]\[
\text{Total All Weeks} = 146 + 155 + 108 = 409
\][/tex]

2. Overall Actual Percentage of Soccer Ball Purchases:
- Add up all the soccer balls sold: [tex]\(85 + 110 + 64 = 259\)[/tex]
- Calculate the overall percentage:
[tex]\[
\text{Overall Actual Percentage} = \frac{259}{409} \approx 0.633
\][/tex]

3. Comparison for Consistency:
- Check if the overall actual percentage is within [tex]\(53\%\)[/tex] to [tex]\(63\%\)[/tex].
- Since [tex]\(63.3\%\)[/tex] is above this range, the total results for all weeks are inconsistent with the model.

### Conclusion:

- Week 1 Results: Consistent with the model.
- Total Results of All 3 Weeks': Inconsistent with the model.

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