Get comprehensive answers to your questions with the help of IDNLearn.com's community. Discover comprehensive answers to your questions from our community of experienced professionals.
Sagot :
To determine if the events "being from Texas" and "preferring brand A" are independent, we need to compare the probability of being from Texas [tex]\( P(\text{Texas}) \)[/tex] with the conditional probability of being from Texas given preferring brand A [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]. If these two probabilities are equal, the events are independent.
Here are the steps to find these probabilities and determine if the events are independent:
### Step 1: Calculate the Probability of Being from Texas [tex]\( P(\text{Texas}) \)[/tex]
The total number of participants is 275.
The number of participants from Texas is 125.
[tex]\[ P(\text{Texas}) = \frac{\text{Number of participants from Texas}}{\text{Total number of participants}} = \frac{125}{275} \approx 0.4545 \][/tex]
### Step 2: Calculate the Probability of Preferring Brand A [tex]\( P(\text{Brand A}) \)[/tex]
The number of participants who prefer Brand A is 176.
[tex]\[ P(\text{Brand A}) = \frac{\text{Number of participants who prefer Brand A}}{\text{Total number of participants}} = \frac{176}{275} \approx 0.64 \][/tex]
### Step 3: Calculate the Conditional Probability of Being from Texas Given Preferring Brand A [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]
The number of participants from Texas who prefer Brand A is 80.
The total number of participants who prefer Brand A is 176.
[tex]\[ P(\text{Texas} \mid \text{Brand A}) = \frac{\text{Number of participants from Texas who prefer Brand A}}{\text{Total number of participants who prefer Brand A}} = \frac{80}{176} \approx 0.4545 \][/tex]
### Step 4: Compare Probabilities to Determine Independence
Compare [tex]\( P(\text{Texas}) \)[/tex] and [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]:
[tex]\[ P(\text{Texas}) = 0.4545 \][/tex]
[tex]\[ P(\text{Texas} \mid \text{Brand A}) = 0.4545 \][/tex]
Since [tex]\( P(\text{Texas}) \approx P(\text{Texas} \mid \text{Brand A}) \)[/tex], the events "being from Texas" and "preferring brand A" are independent.
### Conclusion
The correct answer is:
A. No, they are not independent because [tex]\( P(\text{Texas}) \approx 0.45 \)[/tex] and [tex]\( P(\text{Texas} \mid \text{Brand A}) \approx 0.45 \)[/tex].
Here are the steps to find these probabilities and determine if the events are independent:
### Step 1: Calculate the Probability of Being from Texas [tex]\( P(\text{Texas}) \)[/tex]
The total number of participants is 275.
The number of participants from Texas is 125.
[tex]\[ P(\text{Texas}) = \frac{\text{Number of participants from Texas}}{\text{Total number of participants}} = \frac{125}{275} \approx 0.4545 \][/tex]
### Step 2: Calculate the Probability of Preferring Brand A [tex]\( P(\text{Brand A}) \)[/tex]
The number of participants who prefer Brand A is 176.
[tex]\[ P(\text{Brand A}) = \frac{\text{Number of participants who prefer Brand A}}{\text{Total number of participants}} = \frac{176}{275} \approx 0.64 \][/tex]
### Step 3: Calculate the Conditional Probability of Being from Texas Given Preferring Brand A [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]
The number of participants from Texas who prefer Brand A is 80.
The total number of participants who prefer Brand A is 176.
[tex]\[ P(\text{Texas} \mid \text{Brand A}) = \frac{\text{Number of participants from Texas who prefer Brand A}}{\text{Total number of participants who prefer Brand A}} = \frac{80}{176} \approx 0.4545 \][/tex]
### Step 4: Compare Probabilities to Determine Independence
Compare [tex]\( P(\text{Texas}) \)[/tex] and [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]:
[tex]\[ P(\text{Texas}) = 0.4545 \][/tex]
[tex]\[ P(\text{Texas} \mid \text{Brand A}) = 0.4545 \][/tex]
Since [tex]\( P(\text{Texas}) \approx P(\text{Texas} \mid \text{Brand A}) \)[/tex], the events "being from Texas" and "preferring brand A" are independent.
### Conclusion
The correct answer is:
A. No, they are not independent because [tex]\( P(\text{Texas}) \approx 0.45 \)[/tex] and [tex]\( P(\text{Texas} \mid \text{Brand A}) \approx 0.45 \)[/tex].
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.
