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Select the correct answer.

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

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

A. [tex]\( P \)[/tex] (consumes [tex]\( 1,000-1,500 \)[/tex] calories | weight is 165 lb.) = [tex]\( P \)[/tex] (consumes [tex]\( 1,000-1,500 \)[/tex] calories)

B. [tex]\( P \)[/tex] (weight is 120 lb. | consumes [tex]\( 2,000-2,500 \)[/tex] calories) ≠ [tex]\( P \)[/tex] (weight is 120 lb.)

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

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



Answer :

GinnyAnswer
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]

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