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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.

\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.

\begin{tabular}{|l|l|}
\hline \begin{tabular}{c}
Consistent with \\ Model
\end{tabular} & \begin{tabular}{c}
Inconsistent with \\ Model
\end{tabular} \\
\hline
& \\
& \\
& \\
\hline
\end{tabular}

Answer :

To determine if the results for each category are consistent or inconsistent with the owner's model, we need to compare the percentage of soccer balls sold each week to the expected [tex]$58\%$[/tex] of total sales, with a margin of error of [tex]$\pm 5\%$[/tex].

1. Week 1 Results:
- Soccer Balls: 85
- Baseball Bats: 30
- Tennis Rackets: 31
- Total Sales for Week 1: [tex]$85 + 30 + 31 = 146$[/tex]
- Percentage of Soccer Balls Sold in Week 1: [tex]$\frac{85}{146} \approx 0.5822 \approx 58.22\%$[/tex]

Since [tex]$58.22\%$[/tex] is within the [tex]$53\%$[/tex] to [tex]$63\%$[/tex] range (accounting for the [tex]$\pm 5\%$[/tex] margin), Week 1 is consistent with the model.

2. Week 2 Results:
- Soccer Balls: 110
- Baseball Bats: 22
- Tennis Rackets: 23
- Total Sales for Week 2: [tex]$110 + 22 + 23 = 155$[/tex]
- Percentage of Soccer Balls Sold in Week 2: [tex]$\frac{110}{155} \approx 0.7097 \approx 71.0\%$[/tex]

Since [tex]$71.0\%$[/tex] is outside the [tex]$53\%$[/tex] to [tex]$63\%$[/tex] range, Week 2 is inconsistent with the model.

3. Week 3 Results:
- Soccer Balls: 64
- Baseball Bats: 21
- Tennis Rackets: 23
- Total Sales for Week 3: [tex]$64 + 21 + 23 = 108$[/tex]
- Percentage of Soccer Balls Sold in Week 3: [tex]$\frac{64}{108} \approx 0.5926 \approx 59.26\%$[/tex]

Since [tex]$59.26\%$[/tex] is within the [tex]$53\%$[/tex] to [tex]$63\%$[/tex] range, Week 3 is consistent with the model.

4. Total of All 3 Weeks:
- Total Soccer Balls Sold: [tex]$85 + 110 + 64 = 259$[/tex]
- Total Baseball Bats Sold: [tex]$30 + 22 + 21 = 73$[/tex]
- Total Tennis Rackets Sold: [tex]$31 + 23 + 23 = 77$[/tex]
- Total Sales for All Weeks: [tex]$259 + 73 + 77 = 409$[/tex]
- Percentage of Soccer Balls Sold in All Weeks: [tex]$\frac{259}{409} \approx 0.6330 \approx 63.3\%$[/tex]

Since [tex]$63.3\%$[/tex] is outside the [tex]$53\%$[/tex] to [tex]$63\%$[/tex] range, the total of all 3 weeks is inconsistent with the model.

Therefore, the classification of results according to the model is:

\begin{tabular}{|l|l|}
\hline \begin{tabular}{c}
Consistent with \\
Model
\end{tabular} & \begin{tabular}{c}
Inconsistent with \\
Model
\end{tabular} \\
\hline Week 1 results & Week 2 results \\
Week 3 results & Total of all 3 weeks \\
& \\
\hline
\end{tabular}

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