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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).

\begin{tabular}{|c|c|c|c|c|c|}
\hline Purchase likelihood & 18-34 & [tex]$35-44$[/tex] & [tex]$45-54$[/tex] & [tex]$55+$[/tex] & Total \\
\hline More likely & 207 & 368 & 395 & 408 & 1378 \\
\hline Less likely & 24 & 5 & 25 & 15 & 69 \\
\hline Neither more nor less likely & 297 & 210 & 163 & 108 & 778 \\
\hline Total & 528 & 583 & 583 & 531 & 2225 \\
\hline
\end{tabular}

(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 [tex]$\square$[/tex]

(Round to three decimal places as needed.)



Answer :

GinnyAnswer
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, follow these detailed steps:

1. Identify the relevant values from the contingency table.

The total number of individuals aged 45 to 54 in the survey:
[tex]\[ \text{Total (45-54)} = 583 \][/tex]

The number of individuals aged 45 to 54 who are more likely to buy a product emphasized as "Made in our country":
[tex]\[ \text{More likely (45-54)} = 395 \][/tex]

2. Determine the conditional probability.

The conditional probability [tex]\( P(A|B) \)[/tex] is defined as the probability of event A occurring given that event B has occurred, and it can be calculated using the formula:
[tex]\[ P(\text{More likely}|\text{45-54}) = \frac{\text{Number of individuals more likely (45-54)}}{\text{Total individuals (45-54)}} \][/tex]

3. Substitute the known values into the formula.

[tex]\[ P(\text{More likely}|\text{45-54}) = \frac{395}{583} \][/tex]

4. Calculate the probability.

[tex]\[ P(\text{More likely}|\text{45-54}) \approx 0.6775300171526587 \][/tex]

5. Round the probability to three decimal places.

[tex]\[ P(\text{More likely}|\text{45-54}) \approx 0.678 \][/tex]

Therefore, 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]\( 0.678 \)[/tex].

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