Answer :
Let's carefully analyze each statement from the data in the two-way table and determine which one is true:
### Total Probabilities:
First, we find the totals needed:
- Total weight = 500: The sum of all individuals (180 + 203 + 117).
- Total calories of each range:
- 1000-1500 cal: 140.
- 1500-2000 cal: 250.
- 2000-2500 cal: 110.
- Total by weight:
- 120 lb: 180.
- 145 lb: 203.
- 165 lb: 117.
### Checking Each Statement:
A. [tex]\( P(\text{consumes 1000-1500 calories} \mid \text{weight is 165}) = P(\text{consumes 1000-1500 calories}) \)[/tex]
- Probability of consuming 1000-1500 calories given weight is 165:
[tex]\(\frac{15}{117}\)[/tex].
- Probability of consuming 1000-1500 calories:
[tex]\(\frac{140}{500}\)[/tex].
Since [tex]\(\frac{15}{117} \neq \frac{140}{500}\)[/tex], this statement is false.
B. [tex]\( P(\text{weight is 120 lb} \mid \text{consumes 2000-2500 calories}) \neq P(\text{weight is 120 lb}) \)[/tex]
- Probability of weight being 120 lb given consumes 2000-2500 calories:
[tex]\(\frac{10}{110}\)[/tex].
- Probability of weight being 120 lb:
[tex]\(\frac{180}{500}\)[/tex].
Since [tex]\(\frac{10}{110} \neq \frac{180}{500}\)[/tex], this statement is true.
C. [tex]\( P(\text{weight is 165 lb} \mid \text{consumes 1000-2000 calories}) = P(\text{weight is 165 lb}) \)[/tex]
- Probability of weight being 165 lb given consumes 1000-2000 calories:
[tex]\(\frac{15 + 27}{140 + 250} = \frac{42}{390}\)[/tex].
- Probability of weight being 165 lb:
[tex]\(\frac{117}{500}\)[/tex].
Since [tex]\(\frac{42}{390} \neq \frac{117}{500}\)[/tex], this statement is false.
D. [tex]\( P(\text{weight is 145 lb} \mid \text{consumes 1000-2000 calories}) = P(\text{consumes 1000-2000 calories}) \)[/tex]
- Probability of weight being 145 lb given consumes 1000-2000 calories:
[tex]\(\frac{35 + 143}{140 + 250} = \frac{178}{390}\)[/tex].
- Probability of consuming 1000-2000 calories:
[tex]\(\frac{390}{500}\)[/tex].
Since [tex]\(\frac{178}{390} \neq \frac{390}{500}\)[/tex], this statement is false.
### Conclusion:
Statement B is the only true statement among the four.
### Total Probabilities:
First, we find the totals needed:
- Total weight = 500: The sum of all individuals (180 + 203 + 117).
- Total calories of each range:
- 1000-1500 cal: 140.
- 1500-2000 cal: 250.
- 2000-2500 cal: 110.
- Total by weight:
- 120 lb: 180.
- 145 lb: 203.
- 165 lb: 117.
### Checking Each Statement:
A. [tex]\( P(\text{consumes 1000-1500 calories} \mid \text{weight is 165}) = P(\text{consumes 1000-1500 calories}) \)[/tex]
- Probability of consuming 1000-1500 calories given weight is 165:
[tex]\(\frac{15}{117}\)[/tex].
- Probability of consuming 1000-1500 calories:
[tex]\(\frac{140}{500}\)[/tex].
Since [tex]\(\frac{15}{117} \neq \frac{140}{500}\)[/tex], this statement is false.
B. [tex]\( P(\text{weight is 120 lb} \mid \text{consumes 2000-2500 calories}) \neq P(\text{weight is 120 lb}) \)[/tex]
- Probability of weight being 120 lb given consumes 2000-2500 calories:
[tex]\(\frac{10}{110}\)[/tex].
- Probability of weight being 120 lb:
[tex]\(\frac{180}{500}\)[/tex].
Since [tex]\(\frac{10}{110} \neq \frac{180}{500}\)[/tex], this statement is true.
C. [tex]\( P(\text{weight is 165 lb} \mid \text{consumes 1000-2000 calories}) = P(\text{weight is 165 lb}) \)[/tex]
- Probability of weight being 165 lb given consumes 1000-2000 calories:
[tex]\(\frac{15 + 27}{140 + 250} = \frac{42}{390}\)[/tex].
- Probability of weight being 165 lb:
[tex]\(\frac{117}{500}\)[/tex].
Since [tex]\(\frac{42}{390} \neq \frac{117}{500}\)[/tex], this statement is false.
D. [tex]\( P(\text{weight is 145 lb} \mid \text{consumes 1000-2000 calories}) = P(\text{consumes 1000-2000 calories}) \)[/tex]
- Probability of weight being 145 lb given consumes 1000-2000 calories:
[tex]\(\frac{35 + 143}{140 + 250} = \frac{178}{390}\)[/tex].
- Probability of consuming 1000-2000 calories:
[tex]\(\frac{390}{500}\)[/tex].
Since [tex]\(\frac{178}{390} \neq \frac{390}{500}\)[/tex], this statement is false.
### Conclusion:
Statement B is the only true statement among the four.
