Problem 3 - Basic Statistics for Business and Economics with Student

Question

A total of 1,000 residents in Minnesota were asked which season they preferred. The results were 100 liked winter best, 300 liked spring, 400 liked summer, and 200 liked fall. If the data were summarized in a frequency table, how many classes would be used? What would be the relative frequencies for each class?

Step-by-Step Solution

Answer

4 classes, relative frequencies: winter 0.1, spring 0.3, summer 0.4, fall 0.2.

Step 1: Determine Number of Classes

Each possible season preference represents a different class. Since there are four seasons (winter, spring, summer, fall), there are four classes.

Step 2: Calculate Total Number of Residents

The total number of residents surveyed is 1,000. This represents the entirety of our dataset.

Step 3: Calculate Relative Frequency for Each Season

The relative frequency of a class is calculated as the number of residents who prefer that season divided by the total number of residents surveyed. - For winter: relative frequency = \( \frac{100}{1000} = 0.1 \).- For spring: relative frequency = \( \frac{300}{1000} = 0.3 \).- For summer: relative frequency = \( \frac{400}{1000} = 0.4 \).- For fall: relative frequency = \( \frac{200}{1000} = 0.2 \).

Key Concepts

Relative Frequency

Relative frequency is a helpful concept in statistics, especially when you want to understand how a particular category compares to the total dataset. In the original exercise, relative frequency is used to show what fraction of the survey respondents preferred each season.

It gives us a deeper insight than simple counts because it contextualizes the information. Instead of just knowing 100 people like winter, knowing the relative frequency lets us see this as a 10% preference within the surveyed group. Here's how you can calculate relative frequency:

• You start by dividing the number of occurrences for the category (like's winter preferences) by the total number of occurrences (total survey respondents).
• Formula: \( \text{Relative Frequency} = \frac{\text{Frequency of Class}}{\text{Total Frequency}} \)
• For example, in the survey data: Winter’s relative frequency = \( \frac{100}{1000} = 0.1 \) or 10%.

Relative frequencies are often converted into percentages to effectively communicate results, making them easier to compare at a glance across different categories.

Classes in Statistics

When dealing with frequency distributions, understanding what classes mean is essential. In statistics, classes represent different categories or groups into which data points are divided.

In the context of the exercise, each season represents a class. Here’s a step-by-step of why knowing about classes is vital for analysis:

• Categories or classes help us organize data in a meaningful way, making it easier to interpret results.
• Each class should be exclusive. No data point overlaps into two categories. Thus, in our example, a respondent couldn't prefer both spring and summer simultaneously.
• The number of classes conveys the number of distinct categories being evaluated. In the exercise, with winter, spring, summer, and fall, we have four classes.

Breaking down a dataset into classes allows statisticians and analysts to report on and understand the data's composition quickly. It brings structure and makes the data analysis process more precise.

Survey Analysis

A survey is a powerful research tool that collects quantitative and qualitative data by asking questions to a specific group of people.

Survey analysis involves interpreting the data collected to provide meaningful insights into the topic surveyed. In the given exercise, it’s about understanding Minnesotans' seasonal preferences.

• The first step is to ensure data accuracy by correctly summarizing the number of respondents and their preferences.
• Then, data is usually organized into tables or charts. Our exercise suggested creating a frequency table.
• From this table, calculations like relative frequencies help transform raw data into an understandable format. This helps one make data-driven decisions.
• Survey analysis can inform stakeholders of trends, preferences, and areas of attention needed.

Good survey analysis highlights data patterns and insights that are not immediately apparent, helping guide business decisions, policy-making, and other crucial areas based on factual evidence.