Question

investigating data

ms. johnson uses a statistics program to analyze all the data collected by her students during a lab experiment about acceleration due to gravity. the program reports \\(\overline{x} = 31.5\\) and \\(\sigma = 2.5\\).

what is the variance of the data? round to the nearest tenth.

5.6
6.3
1.6
2.5

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
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"step_cot",
"multiple_choice_formatter",
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"quick"
],
"needs_drawing": false,
"concepts_used": [
"Variance Calculation"
],
"new_concepts": [],
"current_concepts": [
"Standard Deviation",
"Variance Calculation"
]
}
</pre_analysis>

<reasoning>

Identify the given values

\[

$$\begin{aligned} \bar{x} &= 31.5 \\ \sigma &= 2.5 \end{aligned}$$

\]

Apply the variance formula

\[

$$\begin{aligned} \text{Variance} &= \sigma^2 \\ \text{Variance} &= (2.5)^2 \\ \text{Variance} &= 6.25 \end{aligned}$$

\]

Round to the nearest tenth

\[
\text{Variance} \approx 6.3
\]
</reasoning>

<answer>
<mcq-option>5.6</mcq-option>
<mcq-correct>6.3</mcq-correct>
<mcq-option>1.6</mcq-option>
<mcq-option>2.5</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Statistics",
"Variance Calculation"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Variance Calculation"
],
"new_concepts": [],
"current_concepts": [
"Standard Deviation",
"Variance Calculation"
]
}
</pre_analysis>

<reasoning>

Identify the given values

\[

$$\begin{aligned} \bar{x} &= 31.5 \\ \sigma &= 2.5 \end{aligned}$$

\]

Apply the variance formula

\[

$$\begin{aligned} \text{Variance} &= \sigma^2 \\ \text{Variance} &= (2.5)^2 \\ \text{Variance} &= 6.25 \end{aligned}$$

\]

Round to the nearest tenth

\[
\text{Variance} \approx 6.3
\]
</reasoning>

<answer>
<mcq-option>5.6</mcq-option>
<mcq-correct>6.3</mcq-correct>
<mcq-option>1.6</mcq-option>
<mcq-option>2.5</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Statistics",
"Variance Calculation"
]
}
</post_analysis>