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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.

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

Answer :

We start by finding the proportion of customers who purchased soccer balls for each week and then compare these proportions to the model value of [tex]$0.58$[/tex]. A result is considered "consistent" if its soccer ball proportion is within [tex]$0.05$[/tex] (5 percentage points) of [tex]$0.58$[/tex], and "inconsistent" otherwise.

Step 1. Week 1

- Total purchases in Week 1:
[tex]$$85 \,(\text{soccer balls}) + 30 \,(\text{baseball bats}) + 31 \,(\text{tennis rackets}) = 146.$$[/tex]
- Proportion of soccer balls:
[tex]$$\frac{85}{146} \approx 0.582.$$[/tex]
- Comparison:
The absolute difference is
[tex]$$\left|0.582 - 0.58\right| \approx 0.002,$$[/tex]
which is less than [tex]$0.05$[/tex].
Classification: consistent.

Step 2. Week 2

- Total purchases in Week 2:
[tex]$$110 + 22 + 23 = 155.$$[/tex]
- Proportion of soccer balls:
[tex]$$\frac{110}{155} \approx 0.710.$$[/tex]
- Comparison:
The absolute difference is
[tex]$$\left|0.710 - 0.58\right| \approx 0.130,$$[/tex]
which exceeds the [tex]$0.05$[/tex] tolerance.
Classification: inconsistent.

Step 3. Week 3

- Total purchases in Week 3:
[tex]$$64 + 21 + 23 = 108.$$[/tex]
- Proportion of soccer balls:
[tex]$$\frac{64}{108} \approx 0.593.$$[/tex]
- Comparison:
The absolute difference is
[tex]$$\left|0.593 - 0.58\right| \approx 0.013,$$[/tex]
which is within the allowed tolerance of [tex]$0.05$[/tex].
Classification: consistent.

Step 4. Combined (All 3 Weeks)

- Total soccer ball purchases:
[tex]$$85 + 110 + 64 = 259.$$[/tex]
- Total overall purchases:
[tex]$$146 + 155 + 108 = 409.$$[/tex]
- Proportion of soccer balls:
[tex]$$\frac{259}{409} \approx 0.633.$$[/tex]
- Comparison:
The absolute difference is
[tex]$$\left|0.633 - 0.58\right| \approx 0.053,$$[/tex]
which is just over the allowed tolerance of [tex]$0.05$[/tex].
Classification: inconsistent.

Summary of Classifications:

- Total of all 3 weeks' results: Inconsistent
- Week 2 results: Inconsistent
- Week 1 results: Consistent
- Week 3 results: Consistent

This completes the step-by-step solution.

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