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In a recent Harris Poll, a random sample of adult Americans (18 years and older) was asked, "When you see an ad emphasizing that a product is 'Made in America,' 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. $$ \begin{array}{lrrrrr} & \mathbf{1 8 - 3 4} & \mathbf{3 5 - 4 4} & \mathbf{4 5 - 5 4} & \mathbf{5 5 +} & \text { Total } \\ \hline \text { More likely } & 238 & 329 & 360 & 402 & \mathbf{1 3 2 9} \\ \hline \text { Less likely } & 22 & 6 & 22 & 16 & \mathbf{6 6} \\ \hline \begin{array}{l} \text { Neither more } \\ \text { nor less likely } \end{array} & 282 & 201 & 164 & 118 & \mathbf{7 6 5} \\ \hline \text { Total } & \mathbf{5 4 2} & \mathbf{5 3 6} & \mathbf{5 4 6} & \mathbf{5 3 6} & \mathbf{2 1 6 0} \end{array} $$ (a) What is the probability that a randomly selected individual is 35-44 years of age, given the individual is more likely to buy a product emphasized as "Made in America"? (b) What is the probability that a randomly selected individual is more likely to buy a product emphasized as "Made in America," given the individual is \(35-44\) years of age? (c) Are 18 - to 34 -year-olds more likely to buy a product emphasized as "Made in America" than individuals in general?

Step by step solution

01

Understanding the Table

Review the given contingency table. The rows represent the likelihood of buying a product ('More likely', 'Less likely', 'Neither more nor less likely'), while the columns represent age groups (18-34, 35-44, 45-54, 55+). The table provides frequencies for each category and their totals.

02

Probability for 35-44 Years Given 'More Likely'

To find the probability that an individual is 35-44 years old given they are 'more likely' to buy, use the formula for conditional probability, P(A|B) = P(A and B) / P(B). Here, A is the event 'individual is 35-44' and B is the event 'more likely to buy'. Identify P(A and B) = 329 and P(B) = 1329 from the table. Calculate P(35-44 | More Likely) by dividing these values: 329/1329.

03

Probability for 'More Likely' Given 35-44 Years

To find the probability that an individual is 'more likely' to buy given they are 35-44 years old, use P(B|A) = P(A and B) / P(A). Here, B is 'more likely to buy' and A is 'individual is 35-44'. Identify P(A and B) = 329 and P(A) = 536 from the table. Calculate P(More Likely | 35-44) by dividing these values: 329/536.

04

Comparing Likelihood for 18-34 Year Olds with the General Population

Determine if 18-34-year-olds are more likely to buy products over the general population. Calculate the probability for 'more likely to buy' for both 18-34-year-olds and all age groups. For 18-34: P(More Likely | 18-34) = 238/542. For the general population: P(More Likely) = 1329/2160. Compare the two probabilities to make a conclusion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Contingency Table

A contingency table is a type of table in a matrix format that displays the frequency distribution of variables.
It helps in understanding the relationship between two categorical variables by showing their frequencies.
In our exercise, the table evaluates two factors: the age groups (18-34, 35-44, 45-54, and 55+) and their likelihood of buying a product emphasized as 'Made in America'.

The rows show the likelihood categories ('More likely', 'Less likely', 'Neither more nor less likely'), while the columns represent age groups.
Numbers within these cells show the frequency of responses in each age group for each likelihood category.
The totals for each row and column are also provided, which are crucial for various probability calculations.

Understanding how to read and interpret these totals and frequencies allows us to delve deeper into finding conditional probabilities.

Probability Calculations

Probability calculations in this exercise involve determining the likelihood of certain events occurring within the given data.

The core formula used for conditional probability is: \(P(A|B) = \frac{P(A \cap B)}{P(B)} \). \(P(A|B) \) denotes the probability of A occurring given B has occurred, \(P(A \cap B) \) represents the joint probability of A and B occurring, and \(P(B) \) is the probability of B happening alone.

For instance, when asked the probability of being 35-44 years old given an individual is 'more likely' to buy a product emphasized as 'Made in America,' we identify:

• \( P(A \cap B) = 329 \)
• \( P(B) = 1329 \)

Then apply the formula: \(P(35-44|More Likely) = \frac{329}{1329} \approx 0.247 \)

This approach is useful in making probability-based decisions and understanding the dynamics between different categorical variables in our data set.

Age Group Analysis

Age group analysis delves into understanding how different age brackets react to the idea of purchasing a product labeled as 'Made in America'.

Let's break down the probabilities:

• For individuals aged 18-34: To find the probability that they are more likely to buy a product, we look at how often they responded 'More likely': \(P(More Likely|18-34) = \frac{238}{542} \approx 0.439 \).
• For the general population, the probability of someone being 'more likely' to buy a product is: \(P(More Likely) = \frac{1329}{2160} \approx 0.615 \).

By comparing these, we observe that 18-34-year-olds are less likely to purchase 'Made in America' products than the general population.

This analysis helps understand consumer behavior within specific age brackets, aiding in targeted marketing and better product strategy decisions.