Question

the scatter plot shows the time an athlete spends practicing and the number of hours of free time each week.

use technology to write a line of best fit equation and predict the amount of free time, to the nearest hour, for an athlete who practices 9.5 hours per week.

1 hour
3 hours
2 hours
4 hours

Explanation:

<pre_analysis>
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"concepts_used": [
"Scatter Plots",
"Line of Best Fit",
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"current_concepts": [
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</pre_analysis>

<reasoning>

Identify data points from the scatter plot

The coordinates \((x, y)\) representing (Practice, Free Time) are:
\[
(1, 7), (1, 8), (2, 9), (3, 6), (4, 9), (5, 5), (5, 6), (6, 7), (7, 4), (8, 7), (9, 4), (10, 2)
\]

Calculate the line of best fit

Using linear regression formulas:
\[

$$\begin{aligned} n &= 12 \\ \sum x &= 1 + 1 + 2 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 9 + 10 = 61 \\ \sum y &= 7 + 8 + 9 + 6 + 9 + 5 + 6 + 7 + 4 + 7 + 4 + 2 = 74 \\ \sum x^2 &= 1 + 1 + 4 + 9 + 16 + 25 + 25 + 36 + 49 + 64 + 81 + 100 = 411 \\ \sum xy &= 7 + 8 + 18 + 18 + 36 + 25 + 30 + 42 + 28 + 56 + 36 + 20 = 324 \\ m &= \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{12(324) - (61)(74)}{12(411) - (61)^2} = \frac{3888 - 4514}{4932 - 3721} = \frac{-626}{1211} \approx -0.517 \\ b &= \frac{\sum y - m\sum x}{n} = \frac{74 - (-0.517)(61)}{12} \approx \frac{74 + 31.537}{12} \approx 8.795 \end{aligned}$$

\]
The line of best fit equation is:
\[
y \approx -0.517x + 8.795
\]

Predict free time for 9.5 hours of practice

Substitute \(x = 9.5\) into the regression equation:
\[
y \approx -0.517(9.5) + 8.795 \approx -4.9115 + 8.795 = 3.8835
\]
Rounding to the nearest hour gives:
\[
y \approx 4\text{ hours}
\]
</reasoning>

<answer>
<mcq-option>1 hour</mcq-option>
<mcq-option>3 hours</mcq-option>
<mcq-option>2 hours</mcq-option>
<mcq-correct>4 hours</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Statistics",
"Line of Best Fit"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
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</pre_analysis>

<reasoning>

Identify data points from the scatter plot

The coordinates \((x, y)\) representing (Practice, Free Time) are:
\[
(1, 7), (1, 8), (2, 9), (3, 6), (4, 9), (5, 5), (5, 6), (6, 7), (7, 4), (8, 7), (9, 4), (10, 2)
\]

Calculate the line of best fit

Using linear regression formulas:
\[

$$\begin{aligned} n &= 12 \\ \sum x &= 1 + 1 + 2 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 9 + 10 = 61 \\ \sum y &= 7 + 8 + 9 + 6 + 9 + 5 + 6 + 7 + 4 + 7 + 4 + 2 = 74 \\ \sum x^2 &= 1 + 1 + 4 + 9 + 16 + 25 + 25 + 36 + 49 + 64 + 81 + 100 = 411 \\ \sum xy &= 7 + 8 + 18 + 18 + 36 + 25 + 30 + 42 + 28 + 56 + 36 + 20 = 324 \\ m &= \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{12(324) - (61)(74)}{12(411) - (61)^2} = \frac{3888 - 4514}{4932 - 3721} = \frac{-626}{1211} \approx -0.517 \\ b &= \frac{\sum y - m\sum x}{n} = \frac{74 - (-0.517)(61)}{12} \approx \frac{74 + 31.537}{12} \approx 8.795 \end{aligned}$$

\]
The line of best fit equation is:
\[
y \approx -0.517x + 8.795
\]

Predict free time for 9.5 hours of practice

Substitute \(x = 9.5\) into the regression equation:
\[
y \approx -0.517(9.5) + 8.795 \approx -4.9115 + 8.795 = 3.8835
\]
Rounding to the nearest hour gives:
\[
y \approx 4\text{ hours}
\]
</reasoning>

<answer>
<mcq-option>1 hour</mcq-option>
<mcq-option>3 hours</mcq-option>
<mcq-option>2 hours</mcq-option>
<mcq-correct>4 hours</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Statistics",
"Line of Best Fit"
]
}
</post_analysis>