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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.

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

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 results

Answer :

Let's solve the problem systematically and determine if the results for each week and the overall total are consistent with the model (58% of the time, customers buy soccer balls).

Here's the breakdown:

1. Calculate the total number of purchases for each week and overall.
- Week 1: Total purchases = 85 (Soccer Balls) + 30 (Baseball Bats) + 31 (Tennis Rackets) = 146
- Week 2: Total purchases = 110 (Soccer Balls) + 22 (Baseball Bats) + 23 (Tennis Rackets) = 155
- Week 3: Total purchases = 64 (Soccer Balls) + 21 (Baseball Bats) + 23 (Tennis Rackets) = 108
- Overall total for 3 weeks = 146 (Week 1) + 155 (Week 2) + 108 (Week 3) = 409

2. Calculate the total number of soccer balls purchased over three weeks.
- Total soccer balls = 85 (Week 1) + 110 (Week 2) + 64 (Week 3) = 259

3. Calculate the probability of purchasing soccer balls for each week and overall.
- Week 1 probability: [tex]\( \frac{85}{146} \approx 0.582 \)[/tex]
- Week 2 probability: [tex]\( \frac{110}{155} \approx 0.710 \)[/tex]
- Week 3 probability: [tex]\( \frac{64}{108} \approx 0.593 \)[/tex]
- Overall probability: [tex]\( \frac{259}{409} \approx 0.633 \)[/tex]

4. Compare these probabilities with the given model's probability (58%).
- Week 1: [tex]\( 0.582 \approx 58.2\% \)[/tex] (Consistent, because it's slightly above 58%)
- Week 2: [tex]\( 0.710 \approx 71.0\% \)[/tex] (Consistent, because it's well above 58%)
- Week 3: [tex]\( 0.593 \approx 59.3\% \)[/tex] (Consistent, because it's slightly above 58%)
- Overall: [tex]\( 0.633 \approx 63.3\% \)[/tex] (Consistent, because it's above 58%)

Conclusion:
- Week 1 results are consistent with the model.
- Week 2 results are consistent with the model.
- Week 3 results are consistent with the model.
- The overall results for the 3 weeks are consistent with the model.

By breaking down the information above, you can see that in each case, the probability of purchasing soccer balls meets or exceeds the expected 58%, indicating that the results are consistent with the owner’s model.

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Rewritten by : Barada