Let's analyze this step-by-step, focusing on the respective probabilities the question asks for:
(a) What is the probability that a randomly selected individual is 45 to 54 years of age, given the individual is more likely to buy a product emphasized as "Made in our country"?
To find this probability, we use the formula for conditional probability:
[tex]\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \][/tex]
Here:
- [tex]\( A \)[/tex] is the event that an individual is 45 to 54 years of age.
- [tex]\( B \)[/tex] is the event that an individual is more likely to buy a product emphasized as "Made in our country".
From the table:
- The number of individuals who are 45 to 54 years of age and more likely to buy the product is [tex]\( 395 \)[/tex].
- The total number of individuals who are more likely to buy the product is [tex]\( 1378 \)[/tex].
Thus,
[tex]\[ P(\text{45 to 54 years} | \text{More likely}) = \frac{395}{1378} \][/tex]
Using our knowledge, the result is approximately:
[tex]\[ \boxed{0.287} \][/tex]
(b) What is the probability that a randomly selected individual is more likely to buy a product emphasized as "Made in our country," given the individual is 45 to 54 years of age?
Again, using the formula for conditional probability:
[tex]\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \][/tex]
Here:
- [tex]\( A \)[/tex] is the event that an individual is more likely to buy the product.
- [tex]\( B \)[/tex] is the event that an individual is 45 to 54 years of age.
From the table:
- The number of individuals who are 45 to 54 years of age and more likely to buy the product is [tex]\( 395 \)[/tex].
- The total number of individuals who are 45 to 54 years of age is [tex]\( 583 \)[/tex].
Thus,
[tex]\[ P(\text{More likely} | \text{45 to 54 years}) = \frac{395}{583} \][/tex]
Using our knowledge, the probability is approximately:
[tex]\[ \boxed{0.678} \][/tex]
With that detailed analysis, we've found the probabilities for both parts (a) and (b) as follows:
- (a) The probability that a randomly selected individual is 45 to 54 years of age, given the individual is more likely to buy a product emphasized as "Made in our country" is approximately [tex]\( \boxed{0.287} \)[/tex].
- (b) The probability that a randomly selected individual is more likely to buy a product emphasized as "Made in our country," given the individual is 45 to 54 years of age is approximately [tex]\( \boxed{0.678} \)[/tex].
