Questions: In a recent poll, a random sample of adults in some country (18 years and older) was asked, "When you see an ad emphasizing that a product is "Made in our country," are you more likely to buy it, less likely to buy it, or neither more nor less likely to buy it?" The results of the survey, by age group, are presented in the following contingency table. Complete parts (a) through (c). Purchase likelihood 18-3435-4445-5455 + Total More likely 217 332 396 410 1355 Less likely 20 10 29 15 74 Neither more nor less likely 283 203 178 124 788 Total 520 545 603 549 2217 (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"? The probability is approximately 0.292 . (Round to three decimal places as needed.) (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? The probability is approximately 0.657 . (Round to three decimal places as needed.) (c) Are 18 - to 34 -year-olds more likely to buy a product emphasized as "Made in our country' than individuals in general? Yes, more likely No, less likely

Solution Steps

To solve these questions, we need to use conditional probability formulas.

(a) To find 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," we use the formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] where \( A \) is the event that the individual is 45 to 54 years of age, and \( B \) is the event that the individual is more likely to buy the product.

(b) To find 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, we use the formula: \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \] where \( A \) is the event that the individual is 45 to 54 years of age, and \( B \) is the event that the individual is more likely to buy the product.

(c) To determine if 18- to 34-year-olds are more likely to buy a product emphasized as "Made in our country" than individuals in general, we compare the probability of 18- to 34-year-olds being more likely to buy the product to the overall probability of individuals being more likely to buy the product.

To find the probability that a randomly selected individual is 45 to 54 years of age given that the individual is more likely to buy a product emphasized as "Made in our country," we use the formula:

\[ P(45-54 | \text{More likely}) = \frac{P(45-54 \cap \text{More likely})}{P(\text{More likely})} \]

Substituting the values:

\[ P(45-54 | \text{More likely}) = \frac{396}{1355} \approx 0.2923 \]

Thus, rounded to three decimal places:

\[ P(45-54 | \text{More likely}) \approx 0.292 \]

Next, we find the probability that a randomly selected individual is more likely to buy a product emphasized as "Made in our country," given that the individual is 45 to 54 years of age:

\[ P(\text{More likely} | 45-54) = \frac{P(45-54 \cap \text{More likely})}{P(45-54)} \]

Substituting the values:

\[ P(\text{More likely} | 45-54) = \frac{396}{603} \approx 0.6567 \]

Thus, rounded to three decimal places:

\[ P(\text{More likely} | 45-54) \approx 0.657 \]

To determine if 18- to 34-year-olds are more likely to buy a product emphasized as "Made in our country" than individuals in general, we compare:

\[ P(\text{More likely} | 18-34) = \frac{217}{520} \approx 0.4173 \]

and

\[ P(\text{More likely}) = \frac{1355}{2217} \approx 0.6112 \]

Since \( 0.4173 < 0.6112 \), we conclude that 18- to 34-year-olds are less likely to buy the product compared to individuals in general.

Final Answer