Question

g.pr.10.3 (mc)

a study was done on 3300 single men and women to see whether they preferred to live in a house or condominium. the results between gender and housing preference were independent. find the missing values for the males housing preference for house and condo.

\\(\

$$\begin{array}{lccc} & \\text{house} & \\text{condo} & \\text{total} \\\\ \\text{male} & & & 2000 \\\\ \\text{female} & 624 & 676 & 1300 \\\\ \\text{total} & & & 3300 \\end{array}$$

\\)

( ) 1000; 1000
( ) 966; 1034
( ) 624; 676
( ) 960; 1040

Explanation:

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"concepts_used": [
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"new_concepts": [
"Two-Way Frequency Tables"
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"current_concepts": [
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"Two-Way Frequency Tables"
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}
</pre_analysis>

<reasoning>

Calculate the marginal probabilities of housing preference

\[

$$\begin{aligned} P(\text{House} \mid \text{Female}) &= \frac{624}{1300} = 0.48 \\ P(\text{Condo} \mid \text{Female}) &= \frac{676}{1300} = 0.52 \end{aligned}$$

\]

Apply independence condition to males

\[

$$\begin{aligned} P(\text{House} \mid \text{Male}) &= P(\text{House} \mid \text{Female}) = 0.48 \\ P(\text{Condo} \mid \text{Male}) &= P(\text{Condo} \mid \text{Female}) = 0.52 \end{aligned}$$

\]

Compute the missing values for males

\[

$$\begin{aligned} \text{Male House} &= 2000 \times 0.48 = 960 \\ \text{Male Condo} &= 2000 \times 0.52 = 1040 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 1000; 1000</mcq-option>
<mcq-option>(B) 966; 1034</mcq-option>
<mcq-option>(C) 624; 676</mcq-option>
<mcq-correct>(D) 960; 1040</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Statistics",
"Independent Events"
]
}
</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": [
"Independent Events"
],
"new_concepts": [
"Two-Way Frequency Tables"
],
"current_concepts": [
"Independent Events",
"Two-Way Frequency Tables"
]
}
</pre_analysis>

<reasoning>

Calculate the marginal probabilities of housing preference

\[

$$\begin{aligned} P(\text{House} \mid \text{Female}) &= \frac{624}{1300} = 0.48 \\ P(\text{Condo} \mid \text{Female}) &= \frac{676}{1300} = 0.52 \end{aligned}$$

\]

Apply independence condition to males

\[

$$\begin{aligned} P(\text{House} \mid \text{Male}) &= P(\text{House} \mid \text{Female}) = 0.48 \\ P(\text{Condo} \mid \text{Male}) &= P(\text{Condo} \mid \text{Female}) = 0.52 \end{aligned}$$

\]

Compute the missing values for males

\[

$$\begin{aligned} \text{Male House} &= 2000 \times 0.48 = 960 \\ \text{Male Condo} &= 2000 \times 0.52 = 1040 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 1000; 1000</mcq-option>
<mcq-option>(B) 966; 1034</mcq-option>
<mcq-option>(C) 624; 676</mcq-option>
<mcq-correct>(D) 960; 1040</mcq-correct>
</answer>

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