Answer :
To solve this problem, we need to analyze statements about probabilities based on the data in the table. Let's go through each statement one by one.
First, we'll calculate the total number of people and various necessary totals:
- The total population is 500 (the sum of all entries).
- Total for weight 165 lb is 117.
Let's review each statement:
Option A: [tex]\( P \)[/tex] (consumes [tex]\( 1,000-1,500 \)[/tex] calories | weight is 165) = [tex]\( P \)[/tex] (consumes [tex]\( 1,000-1,500 \)[/tex] calories)
1. Probability of consuming 1000-1500 calories given weight is 165 lb:
[tex]\[
P(\text{consumes } 1000-1500 \, \text{calories} | \text{weight is } 165) = \frac{\text{Number of people with 165 lb consuming 1000-1500 calories (15)}}{\text{Total number of people with 165 lb (117)}}
\][/tex]
2. Probability of consuming 1000-1500 calories:
[tex]\[
P(\text{consumes } 1000-1500 \, \text{calories}) = \frac{\text{Total number of people consuming 1000-1500 calories (140)}}{\text{Total population (500)}}
\][/tex]
We compare both probabilities. If they are equal, the statement is true.
Option B: [tex]\( P \)[/tex] (weight is 120 lb | consumes [tex]\( 2,000-2,500 \)[/tex] calories) = [tex]\( P \)[/tex] (weight is 120 lb)
1. Probability of weight being 120 lb given consuming 2000-2500 calories:
[tex]\[
P(\text{weight is 120} | \text{consumes } 2000-2500 \, \text{calories}) = \frac{\text{Number of people with 120 lb consuming 2000-2500 calories (10)}}{\text{Total number of people consuming 2000-2500 calories (110)}}
\][/tex]
2. Probability of weight being 120 lb:
[tex]\[
P(\text{weight is 120}) = \frac{\text{Total number of people with 120 lb (180)}}{\text{Total population (500)}}
\][/tex]
We compare both probabilities. If they are equal, the statement is true.
Option C: [tex]\( P \)[/tex] (weight is 165 lb | consumes [tex]\( 1,000-2,000 \)[/tex] calories) = [tex]\( P \)[/tex] (weight is 165 lb)
1. Probability of weight being 165 lb given consuming 1000-2000 calories:
[tex]\[
P(\text{weight is 165} | \text{consumes } 1000-2000 \, \text{calories}) = \frac{\text{Number of people with 165 lb consuming 1000-2000 calories (15+27)}}{\text{Total number of people with 165 lb (117)}}
\][/tex]
2. Probability of weight being 165 lb:
[tex]\[
P(\text{weight is 165}) = \frac{\text{Total number of people with 165 lb (117)}}{\text{Total population (500)}}
\][/tex]
We compare both probabilities. If they are equal, the statement is true.
Option 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)
1. Probability of weight being 145 lb given consuming 1000-2000 calories:
[tex]\[
P(\text{weight is 145} | \text{consumes } 1000-2000 \, \text{calories}) = \frac{\text{Number of people with 145 lb consuming 1000-2000 calories (35+143)}}{\text{Total number of people consuming 1000-2000 calories (250)}}
\][/tex]
2. Probability of consuming 1000-2000 calories:
[tex]\[
P(\text{consumes } 1000-2000 \, \text{calories}) = \frac{\text{Total number of people consuming 1000-2000 calories (250)}}{\text{Total population (500)}}
\][/tex]
We compare both probabilities. If they are equal, the statement is true.
After evaluating, we find that none of the statements are true. Therefore, the correct answer is that none of the statements hold, which corresponds to "None."
First, we'll calculate the total number of people and various necessary totals:
- The total population is 500 (the sum of all entries).
- Total for weight 165 lb is 117.
Let's review each statement:
Option A: [tex]\( P \)[/tex] (consumes [tex]\( 1,000-1,500 \)[/tex] calories | weight is 165) = [tex]\( P \)[/tex] (consumes [tex]\( 1,000-1,500 \)[/tex] calories)
1. Probability of consuming 1000-1500 calories given weight is 165 lb:
[tex]\[
P(\text{consumes } 1000-1500 \, \text{calories} | \text{weight is } 165) = \frac{\text{Number of people with 165 lb consuming 1000-1500 calories (15)}}{\text{Total number of people with 165 lb (117)}}
\][/tex]
2. Probability of consuming 1000-1500 calories:
[tex]\[
P(\text{consumes } 1000-1500 \, \text{calories}) = \frac{\text{Total number of people consuming 1000-1500 calories (140)}}{\text{Total population (500)}}
\][/tex]
We compare both probabilities. If they are equal, the statement is true.
Option B: [tex]\( P \)[/tex] (weight is 120 lb | consumes [tex]\( 2,000-2,500 \)[/tex] calories) = [tex]\( P \)[/tex] (weight is 120 lb)
1. Probability of weight being 120 lb given consuming 2000-2500 calories:
[tex]\[
P(\text{weight is 120} | \text{consumes } 2000-2500 \, \text{calories}) = \frac{\text{Number of people with 120 lb consuming 2000-2500 calories (10)}}{\text{Total number of people consuming 2000-2500 calories (110)}}
\][/tex]
2. Probability of weight being 120 lb:
[tex]\[
P(\text{weight is 120}) = \frac{\text{Total number of people with 120 lb (180)}}{\text{Total population (500)}}
\][/tex]
We compare both probabilities. If they are equal, the statement is true.
Option C: [tex]\( P \)[/tex] (weight is 165 lb | consumes [tex]\( 1,000-2,000 \)[/tex] calories) = [tex]\( P \)[/tex] (weight is 165 lb)
1. Probability of weight being 165 lb given consuming 1000-2000 calories:
[tex]\[
P(\text{weight is 165} | \text{consumes } 1000-2000 \, \text{calories}) = \frac{\text{Number of people with 165 lb consuming 1000-2000 calories (15+27)}}{\text{Total number of people with 165 lb (117)}}
\][/tex]
2. Probability of weight being 165 lb:
[tex]\[
P(\text{weight is 165}) = \frac{\text{Total number of people with 165 lb (117)}}{\text{Total population (500)}}
\][/tex]
We compare both probabilities. If they are equal, the statement is true.
Option 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)
1. Probability of weight being 145 lb given consuming 1000-2000 calories:
[tex]\[
P(\text{weight is 145} | \text{consumes } 1000-2000 \, \text{calories}) = \frac{\text{Number of people with 145 lb consuming 1000-2000 calories (35+143)}}{\text{Total number of people consuming 1000-2000 calories (250)}}
\][/tex]
2. Probability of consuming 1000-2000 calories:
[tex]\[
P(\text{consumes } 1000-2000 \, \text{calories}) = \frac{\text{Total number of people consuming 1000-2000 calories (250)}}{\text{Total population (500)}}
\][/tex]
We compare both probabilities. If they are equal, the statement is true.
After evaluating, we find that none of the statements are true. Therefore, the correct answer is that none of the statements hold, which corresponds to "None."
