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Sagot :
To determine which statement is true, we need to compare the probabilities given in the statements. Let's start by calculating the relevant probabilities.
### Given Data:
- Total number of people = 500
- Breakdown by weight:
- 120 lbs: 180 people
- 145 lbs: 203 people
- 165 lbs: 117 people
- Breakdown by calories:
- 1000-1500 calories: 140 people
- 1500-2000 calories: 250 people
- 2000-2500 calories: 110 people
### Statement A: [tex]\( P(\text{consumes 1000-1500 calories} | \text{weight is 165}) = P(\text{consumes 1000-1500 calories}) \)[/tex]
- [tex]\( P(\text{weight is 165}) = \frac{117}{500} \)[/tex]
- [tex]\( P(\text{consumes 1000-1500 calories} | \text{weight is 165}) = \frac{15}{117} \)[/tex]
- [tex]\( P(\text{consumes 1000-1500 calories}) = \frac{140}{500} \)[/tex]
Comparing these values, they are not equal:
[tex]\[ \frac{15}{117} \neq \frac{140}{500} \][/tex]
### Statement B: [tex]\( P(\text{weight is 120 lbs} \cap \text{consumes 2000-2500 calories}) \neq P(\text{weight is 120 lbs}) \)[/tex]
- [tex]\( P(\text{weight is 120 lbs}) = \frac{180}{500} \)[/tex]
- [tex]\( P(\text{weight is 120 lbs} \cap \text{consumes 2000-2500 calories}) = \frac{10}{500} \)[/tex]
Comparing these values, they are not equal:
[tex]\[ \frac{10}{500} \neq \frac{180}{500} \][/tex]
### Statement C: [tex]\( P(\text{weight is 165 lbs} \cap \text{consumes 1000-2000 calories}) = P(\text{weight is 165 lbs}) \)[/tex]
- [tex]\( P(\text{weight is 165 lbs}) = \frac{117}{500} \)[/tex]
- [tex]\( P(\text{weight is 165 lbs} \cap \text{consumes 1000-2000 calories}) = \frac{15 + 27}{500} = \frac{42}{500} \)[/tex]
Comparing these values, they are not equal:
[tex]\[ \frac{42}{500} \neq \frac{117}{500} \][/tex]
### Statement D: [tex]\( P(\text{weight is 145 lbs and consumes 1000-2000 calories}) = P(\text{consumes 1000-2000 calories}) \)[/tex]
- [tex]\( P(\text{weight is 145 lbs and consumes 1000-2000 calories}) = \frac{35 + 143}{500} = \frac{178}{500} \)[/tex]
- [tex]\( P(\text{consumes 1000-2000 calories}) = \frac{140 + 250}{500} = \frac{390}{500} \)[/tex]
Comparing these values, they are not equal:
[tex]\[ \frac{178}{500} \neq \frac{390}{500} \][/tex]
Based on these calculations, we find that only Statement B is true. Therefore, the correct answer is:
### Statement B. [tex]\( P(\text{weight is 120 lbs} \cap \text{consumes 2000-2500 calories}) \neq P(\text{weight is 120 lbs}) \)[/tex]
### Given Data:
- Total number of people = 500
- Breakdown by weight:
- 120 lbs: 180 people
- 145 lbs: 203 people
- 165 lbs: 117 people
- Breakdown by calories:
- 1000-1500 calories: 140 people
- 1500-2000 calories: 250 people
- 2000-2500 calories: 110 people
### Statement A: [tex]\( P(\text{consumes 1000-1500 calories} | \text{weight is 165}) = P(\text{consumes 1000-1500 calories}) \)[/tex]
- [tex]\( P(\text{weight is 165}) = \frac{117}{500} \)[/tex]
- [tex]\( P(\text{consumes 1000-1500 calories} | \text{weight is 165}) = \frac{15}{117} \)[/tex]
- [tex]\( P(\text{consumes 1000-1500 calories}) = \frac{140}{500} \)[/tex]
Comparing these values, they are not equal:
[tex]\[ \frac{15}{117} \neq \frac{140}{500} \][/tex]
### Statement B: [tex]\( P(\text{weight is 120 lbs} \cap \text{consumes 2000-2500 calories}) \neq P(\text{weight is 120 lbs}) \)[/tex]
- [tex]\( P(\text{weight is 120 lbs}) = \frac{180}{500} \)[/tex]
- [tex]\( P(\text{weight is 120 lbs} \cap \text{consumes 2000-2500 calories}) = \frac{10}{500} \)[/tex]
Comparing these values, they are not equal:
[tex]\[ \frac{10}{500} \neq \frac{180}{500} \][/tex]
### Statement C: [tex]\( P(\text{weight is 165 lbs} \cap \text{consumes 1000-2000 calories}) = P(\text{weight is 165 lbs}) \)[/tex]
- [tex]\( P(\text{weight is 165 lbs}) = \frac{117}{500} \)[/tex]
- [tex]\( P(\text{weight is 165 lbs} \cap \text{consumes 1000-2000 calories}) = \frac{15 + 27}{500} = \frac{42}{500} \)[/tex]
Comparing these values, they are not equal:
[tex]\[ \frac{42}{500} \neq \frac{117}{500} \][/tex]
### Statement D: [tex]\( P(\text{weight is 145 lbs and consumes 1000-2000 calories}) = P(\text{consumes 1000-2000 calories}) \)[/tex]
- [tex]\( P(\text{weight is 145 lbs and consumes 1000-2000 calories}) = \frac{35 + 143}{500} = \frac{178}{500} \)[/tex]
- [tex]\( P(\text{consumes 1000-2000 calories}) = \frac{140 + 250}{500} = \frac{390}{500} \)[/tex]
Comparing these values, they are not equal:
[tex]\[ \frac{178}{500} \neq \frac{390}{500} \][/tex]
Based on these calculations, we find that only Statement B is true. Therefore, the correct answer is:
### Statement B. [tex]\( P(\text{weight is 120 lbs} \cap \text{consumes 2000-2500 calories}) \neq P(\text{weight is 120 lbs}) \)[/tex]
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