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[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{Weight/Calories per Day} & \text{1000 to 1500 cal.} & \text{1500 to 2000 cal.} & \text{2000 to 2500 cal.} & \text{Total} \\
\hline
\text{120 lb.} & 90 & 80 & 10 & 180 \\
\hline
\text{145 lb.} & 35 & 143 & 25 & 203 \\
\hline
\text{165 lb.} & 15 & 27 & 75 & 117 \\
\hline
\text{Total} & 140 & 250 & 110 & 500 \\
\hline
\end{array}
\][/tex]

Based on the data in the two-way table, which statement is true?

A. [tex]\( P(\text{consumes } 1,000-1,500 \text{ calories} \mid \text{weight is 165}) = P(\text{consumes } 1,000-1,500 \text{ calories}) \)[/tex]

B. [tex]\( P(\text{weight is 120 lb.} \mid \text{consumes } 2,000-2,500 \text{ calories}) \neq P(\text{weight is 120 lb.}) \)[/tex]

C. [tex]\( P(\text{weight is 165 lb. consumes } 1,000-2,000 \text{ calories}) = P(\text{weight is 165 lb.}) \)[/tex]

D. [tex]\( P(\text{weight is 145 lb.} \mid \text{consumes } 1,000-2,000 \text{ calories}) = P(\text{consumes } 1,000-2,000 \text{ calories}) \)[/tex]

Answer :

To solve this problem, we need to compare the probabilities given in the statements. Let's evaluate each option in the context of the data provided.

### Two-way table data summary:

- Total participants: 500

Weight Categories:

- 120 lb.: Total 180 participants
- 1000 to 1500 cal: 90
- 1500 to 2000 cal: 80
- 2000 to 2500 cal: 10

- 145 lb.: Total 203 participants
- 1000 to 1500 cal: 35
- 1500 to 2000 cal: 143
- 2000 to 2500 cal: 25

- 165 lb.: Total 117 participants
- 1000 to 1500 cal: 15
- 1500 to 2000 cal: 27
- 2000 to 2500 cal: 75

Calorie Categories:

- 1000 to 1500 cal: Total 140 participants
- 1500 to 2000 cal: Total 250 participants
- 2000 to 2500 cal: Total 110 participants

### Evaluating each statement:

Option A:
[tex]\( P \)[/tex] (consumes 1000-1500 calories | weight is 165) [tex]\( = P \)[/tex] (consumes 1000-1500 calories)

- Probability a person weighing 165 lb consumes 1000-1500 calories:
[tex]\[ \frac{15}{117} = 0.1282 \][/tex]

- Probability a person consumes 1000-1500 calories:
[tex]\[ \frac{140}{500} = 0.28 \][/tex]

Since these probabilities are not equal, Option A is false.

Option B:
[tex]\( P \)[/tex] (weight is 120 lb | consumes 2000-2500 calories) [tex]\( \neq P \)[/tex] (weight is 120 lb)

- Probability a person is 120 lb given they consume 2000-2500 calories:
[tex]\[ \frac{10}{110} = 0.0909 \][/tex]

- Probability a person weighs 120 lb:
[tex]\[ \frac{180}{500} = 0.36 \][/tex]

These probabilities are not equal, so Option B is true.

Option C:
[tex]\( P \)[/tex] (weight is 165 lb and consumes 1000-2000 calories) [tex]\( = P \)[/tex] (weight is 165 lb)

- Probability a person weighs 165 lb and consumes 1000-2000 calories:
[tex]\[ \frac{15 + 27}{500} = 0.084 \][/tex]

- Probability a person weighs 165 lb:
[tex]\[ \frac{117}{500} = 0.234 \][/tex]

Since these are not equal, Option C is false.

Option D:
[tex]\( P \)[/tex] (weight is 145 lb | consumes 1000-2000 calories) [tex]\( = P \)[/tex] (consumes 1000-2000 calories)

- Probability a person is 145 lb given they consume 1000-2000 calories:
[tex]\[ \frac{35 + 143}{140 + 250} = 0.4564 \][/tex]

- Probability a person consumes 1000-2000 calories:
[tex]\[ \frac{140 + 250}{500} = 0.78 \][/tex]

As they are not equal, Option D is false.

### Conclusion:

Based on the evaluations, the true statement is Option B.

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