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.
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.
