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Chapter 3: Problem 25

A recent survey of a new diet cola reported the following percentages of people who liked the taste. Find the weighted mean of the percentages. \(\begin{array}{ccc}{\text { Area }} & {\% \text { Favored }} & {\text { Number surveyed }} \\ \hline 1 & {40} & {1000} \\ {2} & {30} & {3000} \\ {3} & {50} & {800}\end{array}\)

Short Answer

Expert verified
The weighted mean percentage is approximately 35.42%.

Step by step solution

01

Understanding Weighted Mean

A weighted mean considers the weight of each value, which in this case is the number of people surveyed in each area. The formula for weighted mean is \( \bar{x}_w = \frac{\sum (x_i \cdot w_i)}{\sum w_i} \), where \( x_i \) is the percentage liked and \( w_i \) is the number surveyed.
02

Calculate Weighted Sums

For each area, multiply the percentage of people who liked the taste by the number of people surveyed in that area. - Area 1: \( 40 \times 1000 = 40,000 \)- Area 2: \( 30 \times 3000 = 90,000 \)- Area 3: \( 50 \times 800 = 40,000 \)
03

Sum of Weights and Weighted Sums

Add the numbers surveyed to find the total weight: \(1000 + 3000 + 800 = 4800\). Similarly, sum the weighted values calculated earlier:\(40,000 + 90,000 + 40,000 = 170,000\).
04

Calculate the Weighted Mean

Apply the weighted mean formula: \(\bar{x}_w = \frac{170,000}{4800} \approx 35.42\).
05

Interpret the Result

The weighted mean percentage of people who liked the taste of the cola across all areas is approximately 35.42%.

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Key Concepts

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

Statistics
Statistics is all about collecting, analyzing, interpreting, and presenting data. It helps in making sense of raw data by giving it a meaningful structure. When it comes to surveys, statistics can help us derive insights from people's responses. Key activities in statistics include:
  • Collecting data: Gathering information through surveys and experiments.
  • Organizing data: Sorting the responses and organizing them in a visible format, like tables or charts.
  • Analyzing data: Applying mathematical procedures to summarize or describe the data patterns.
  • Drawing conclusions: Understanding the underlying trends to make informed decisions.
This example with the diet cola survey illustrates the application of statistics to understand public opinion. Statistics empowers us to turn basic figures into information on which companies and individuals can base their decisions.
Survey Analysis
Survey analysis focuses on making sense of data collected from surveys. It involves different processes to interpret people's responses. In our cola taste example, survey analysis begins by identifying key areas to investigate, such as the different geographic regions where the survey took place.
In a survey-driven analysis, you need to:
  • Define objectives: Know what you want to learn from the survey.
  • Design the survey: Develop questions that align with the objectives.
  • Collect the data: Engage participants to answer the questions.
  • Analyze the responses: Use statistical methods to examine the data.
The ultimate aim of survey analysis is to uncover meaningful patterns, relationships, or trends. It presents the collated data in a form that can guide strategic decisions or actions.
Weighted Average
The weighted average is a type of mean that takes into account the importance or frequency of each value in a data set. Unlike a simple average, where each item is considered equal, a weighted average assigns weights to values based on their significance.
In our survey, the weight is the number of people surveyed in each region. The formula used is:\[ \bar{x}_w = \frac{\sum (x_i \cdot w_i)}{\sum w_i} \]where:
  • \(x_i\) is each percentage value of favorability.
  • \(w_i\) is the number of people surveyed.
Here, weights help us understand the average percentage of people who liked the cola, reflecting the contribution of each surveyed group. Weighted averages provide a balanced mean that properly reflects varied group sizes or importance.
Data Interpretation
Data interpretation involves examining data to draw conclusions and make informed choices. In the context of the cola survey, data interpretation helped us determine how much people liked the new diet cola in different regions.
Interpreting the weighted mean of 35.42% indicates the combined likeness level, considering survey contributions across all areas. It goes beyond the obvious numbers, using statistical methods to present a clearer image of overall trends.
To effectively interpret data, one should:
  • Understand the context: Know why and how the data was collected.
  • Apply mathematical formulas: Use appropriate tools to analyze the data.
  • Draw conclusions: Determine what the numbers mean in real-world settings.
  • Communicate findings: Present information clearly for a broader audience.
With thoughtful data interpretation, surveys like this provide valuable insights, facilitating smarter decisions and understanding audience sentiments.

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