91Ó°ÊÓ

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 contingency table below. 3 $$\begin{array}{lrrrrr} & 18-34 & 35-44 & 45-54 & 55+ & \text { Total } \\ \hline \text { More likely } & 238 & 329 & 360 & 402 & 1329 \\\\\hline \text { Less likely } & 22 & 6 & 22 & 16 & 66 \\\\\hline \begin{array}{l}\text { Neither more } \\\\\text { nor less likely }\end{array} & 282 & 201 & 164 & 118 & 765 \\\\\hline \text { Total } & 542 & 536 & 546 & 536 & 2160\end{array}$$ (a) How many adult Americans were surveyed? How many were 55 and older? (b) Construct a relative frequency marginal distribution. (c) What proportion of Americans are more likely to buy a product when the ad says "Made in America"? (d) Construct a conditional distribution of likelihood to buy "Made in America" by age. That is, construct a conditional distribution treating age as the explanatory variable. (e) Draw a bar graph of the conditional distribution found in part (d). (f) Write a couple sentences explaining any relation between likelihood to buy and age.

Step by step solution

01

Determine Total Number of Surveyed Adult Americans

Sum the total number of individuals surveyed from all age groups which is provided in the last row of the table.

02

Determine Number of Adults 55 and Older

Identify the total number of adults in the '55+' age group from the table.

03

Construct Relative Frequency Marginal Distribution

Calculate the relative frequencies for each category by dividing the total number of responses in each category by the overall total number of respondents. Express these frequencies as percentages.

04

Calculate Proportion of Americans More Likely to Buy

Divide the number of respondents who are 'more likely to buy' by the total number of respondents and express it as a proportion or percentage.

05

Construct Conditional Distribution by Age

For each age group, calculate the number of respondents in each likelihood to buy category ('more likely', 'less likely', 'neither more nor less likely') as a percentage of the total number of respondents in that age group.

06

Draw Bar Graph

Plot a bar graph for the conditional distribution from Step 5. The x-axis should represent the age groups, and the y-axis should represent the percentage of responses in each likelihood to buy category.

07

Analyze the Relation between Likelihood to Buy and Age

Observe the bar graph and write a couple of sentences discussing any trends or relationships between age and the likelihood to buy products advertised as 'Made in America'.

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Key Concepts

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

Relative Frequency Distribution

A relative frequency distribution helps you understand the proportion of observations within different categories compared to the total number. To construct this, you divide the number of responses in each category by the overall total and express it as a percentage.
For instance, in our survey, we can calculate the relative frequency for 'More likely' by taking the total number of people who are more likely to buy (1329) and dividing it by the overall survey total (2160). This results in a relative frequency of approximately 61.53%.
This type of distribution provides insight into how common each response is relative to the others. It's extremely useful for summarizing large datasets succinctly.

Conditional Distribution

Conditional distribution helps in understanding the probability of an event given a particular condition. In the context of our survey, it involves breaking down responses by age group to see how each age group responded to the advertising based on 'Made in America'.
To construct this, we calculate the proportion of each likelihood category (more likely, less likely, neither more nor less likely) within each age group.
For example, for the 18-34 age group, you will divide the number of 'More likely' responses (238) by the total in that age group (542). This process is repeated for each category within each age group to get the conditional distributions.

Survey Analysis

Conducting a survey analysis involves collecting and analyzing responses to gather data-driven insights. Surveys like the one discussed can reveal behavioral trends and preferences among different demographics.
The first step involves data collection, which in this exercise, includes responses from adult Americans on whether they are more likely, less likely, or neither likely to buy products advertised as 'Made in America.'
After collecting the data, analyzing it using contingency tables, relative frequencies, and conditional distributions can uncover important trends, such as which age groups are more influenced by 'Made in America' advertising.

Bar Graph

Bar graphs are a visual tool to represent data, making it easier to identify trends and comparisons. In our case, a bar graph helps visualize the conditional distribution of likelihood to buy by age group.
The x-axis typically shows the age groups (like 18-34, 35-44, etc.), while the y-axis shows the percentage of responses in each likelihood category.
Each category (More likely, Less likely, Neither more nor less likely) is represented by different colored bars for comparison within the age groups. This makes it simple to compare how different age groups responded to the survey question.

Marginal Distribution

Marginal distribution focuses on the totals of each category, ignoring other variables. It is derived directly from the totals in contingency tables.
For instance, looking at 'More likely' responses, the marginal distribution would involve examining the total number of people who selected 'More likely' across all age groups.
This can be expressed as a proportion of the total respondents, allowing us to understand the overall trend independently of age groups.
Calculating these helps assess the general tendency without the influence of other variables, offering a broad overview of the data trends.