High School

Select the correct answer.

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

Let's analyze each statement based on the data provided.

First, let's explain what these probability expressions represent:

- Conditional Probability: This is the probability of an event occurring given that another event has occurred. It is represented as [tex]\( P(A \mid B) \)[/tex], meaning the probability of [tex]\( A \)[/tex] given [tex]\( B \)[/tex].

Now, let's break down each statement:

Statement A:
- [tex]\( P(\text{consumes } 1,000-1,500 \text{ calories} \mid \text{weight is 165}) \)[/tex]
- This is the probability of someone consuming 1,000-1,500 calories given they weigh 165 lb.
- From the table: 15 people weigh 165 lb. and consume 1,000-1,500 calories, and a total of 117 people weigh 165 lb.
- Probability: [tex]\( \frac{15}{117} \)[/tex].

- [tex]\( P(\text{consumes } 1,000-1,500 \text{ calories}) \)[/tex]
- This is the probability of someone consuming 1,000-1,500 calories overall.
- From the table: 140 people consume 1,000-1,500 calories out of a total of 500 people.
- Probability: [tex]\( \frac{140}{500} \)[/tex].

Since these two probabilities do not match, statement A is false.

Statement B:
- [tex]\( P(\text{weight is } 120 \text{ lb} \mid \text{consumes } 2,000-2,500 \text{ calories}) \)[/tex]
- This is the probability of someone being 120 lb given they consume 2,000-2,500 calories.
- From the table: 10 people consume 2,000-2,500 calories with a weight of 120 lb. The total number of people consuming 2,000-2,500 calories is 110.
- Probability: [tex]\( \frac{10}{110} \)[/tex].

- [tex]\( P(\text{weight is } 120 \text{ lb}) \)[/tex]
- This is the probability of someone weighing 120 lb regardless of calorie consumption.
- From the table: 180 people weigh 120 lb out of a total of 500 people.
- Probability: [tex]\( \frac{180}{500} \)[/tex].

Since these probabilities are not equal, statement B is true.

Statement C:
- [tex]\( P(\text{weight is 165 lb consumes } 1,000-2,000 \text{ calories}) \)[/tex]
- Here, we combine the probabilities for calories 1,000-1,500 and 1,500-2,000.
- From the table: 15 (for 1,000-1,500) + 27 (for 1,500-2,000) = 42 people fit both conditions, with a total weight category of 165 lb.
- Probability (given we're considering only those who weigh 165 lb): [tex]\( \frac{42}{117} \)[/tex].

- [tex]\( P(\text{weight is 165 lb}) \)[/tex]
- As calculated before, the probability is [tex]\( \frac{117}{500} \)[/tex].

These probabilities are not equal, so statement C is false.

Statement D:
- [tex]\( P(\text{weight is 145 lb} \mid \text{consumes } 1,000-2,000 \text{ calories}) \)[/tex]
- From the table for consuming 1,000-2,000 calories: 35 (for 1,000-1,500) + 143 (for 1,500-2,000) = 178 people with weight 145 lb.
- Total consuming these calories is 140 + 250 = 390 people.
- Probability: [tex]\( \frac{178}{390} \)[/tex].

- [tex]\( P(\text{consumes } 1,000-2,000 \text{ calories}) \)[/tex]
- Calculated total consuming 1,000-2,000 calories out of the total 500 people.
- Probability: [tex]\( \frac{390}{500} \)[/tex].

These probabilities are not equal, so statement D is false.

The correct statement after analyzing the data is Statement B.