91Ó°ÊÓ

A July 2011 ESPN SportsNation poll asked, "Which is the best Fourth of July weekend sports tradition?" (http://espn.go.com/espn/fp/flashPollResultsState?sportIndex=frontpage\&pollid=116290). The choices were Major League baseball game (B), Nathan's Famous International Hot Dog Eating Contest (H), Breakfast at Wimbledon (W), or NASCAR race at Daytona (N). The following data represent the responses of a random sample of 45 persons who were asked the same question. \(\begin{array}{lllllllll}\text { H } & \text { H } & \text { B } & \text { W } & \text { N } & \text { B } & \text { H } & \text { N } & \text { W } \\\ \text { N } & \text { H } & \text { B } & \text { W } & \text { H } & \text { N } & \text { N } & \text { H } & \text { H } \\ \text { B } & \text { B } & \text { W } & \text { H } & \text { H } & \text { B } & \text { W } & \text { H } & \text { B } \\ \text { H } & \text { B } & \text { B } & \text { H } & \text { B } & \text { H } & \text { B } & \text { N } & \text { H } \\ \text { B } & \text { B } & \text { H } & \text { H } & \text { H } & \text { B } & \text { H } & \text { H } & \text { N }\end{array}\) a. Prepare a frequency distribution table. b. Calculate the relative frequencies and percentages for all categories. c. What percentage of the respondents mentioned Major League baseball game or Breakfast at Wimbledon? d. Draw a bar graph for the frequency distribution.

Step by step solution

01

Preparation of Frequency Distribution Table

List down all the unique categories of the data i.e. B, H, W, N. Under these categories, count the occurrence of each category in the provided data. This forms the frequency distribution.

02

Calculation of Relative Frequencies and Percentages

Relative frequency for a category is calculated by dividing the frequency of the category by the total number of data points. To convert this into percentage, multiply the relative frequency by 100. Apply this method for each category.

03

Calculating Percentage for B and W

Add the relative frequencies for categories B and W obtained in Step 2, and multiply with 100 to get the percentage of respondents mentioning either Major League Baseball game (B) or Breakfast at Wimbledon (W).

04

Constructing a Bar Graph

On a graph paper or using any graph-plotting tool, label the x-axis with the names of categories and y-axis with the frequencies. Draw bars of heights corresponding to the frequency of each category. This is the frequency distribution graph.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Relative Frequencies

Relative frequencies are a helpful tool in statistics that allow us to understand the ratio of a particular event occurring in a data set compared to the total number of events. To calculate relative frequencies, use the formula:

• Relative Frequency = \( \frac{\text{Frequency of Category}}{\text{Total Number of Data Points}} \)

When analyzing survey data, we often convert these relative frequencies into percentages to make the information more digestible. Multiplying by 100 provides a straightforward interpretation as a percentage. This helps in comparing different categories within the data. For example, in the sports tradition poll data:

• "H" might appear 18 times out of 45 total responses, giving a relative frequency of \( \frac{18}{45} \approx 0.4 \).
• As a percentage, this is \( 0.4 \times 100 = 40\% \).

Thus, relative frequencies and their percentages allow quick insights into the prevalence of each category within the dataset.

Bar Graph

A bar graph is a visual representation of data that uses rectangular bars to show the frequency or proportion of different categories. Each bar's height corresponds to the category's frequency, making it easy to compare across categories. When creating a bar graph from survey data:

• Select appropriate labels for the x-axis (categories like B, H, W, N).
• Mark the y-axis with increments of frequency or percentage.
• Draw bars for each category to the corresponding height.

Bar graphs offer an immediate visual comparison and are especially useful when addressing questions like which category was most or least popular. In our poll example, the height of the bar for "H" would be quite stark if it were the most selected option, allowing students to take in the information quickly.

Percentage Calculation

Percentage calculation is an essential math skill used to express a number as a fraction of 100. This is especially useful in interpreting survey data, as it provides context about the proportion of the sample favoring a particular choice. To find the percentage:

• First, obtain the relative frequency of a category by dividing its frequency by the total number of responses.
• Multiply this relative frequency by 100 to convert it to a percentage.

For instance, if the frequency of voters preferring Major League Baseball game "B" was 16 out of 45:

• Relative frequency = \( \frac{16}{45} \approx 0.356 \).
• Percentage = \( 0.356 \times 100 \approx 35.6\% \).

Calculating percentages helps clarify how dominant or minor a particular opinion is within the group survey, ensuring a more accessible and interpretative result.

Survey Data Analysis

Survey data analysis involves examining the responses collected from surveys to uncover important insights and trends. This often includes creating frequency distributions, calculating relative frequencies and percentages, and utilizing visual aids like bar graphs. Each step in analysis:

• Organizes the raw data in a way that highlights key patterns.
• Facilitates a deeper understanding of preferences and behaviors of respondents.
• Transforms qualitative feedback into quantitative insights.

For the given exercise, for example, we identify how many people chose each sports tradition. Calculations reveal the relative popularity, and visualization through a bar graph enhances engagement. Survey analysis gives context to numbers, helping to make informed decisions or crafting conclusions based on gathered data.