Problem 4 - Basic Statistics for Business and Economics with Student

Question

Two thousand frequent Midwestern business travelers are asked which Midwest city they prefer: Indianapolis, Saint Louis, Chicago, or Milwaukee. The results were 100 liked Indianapolis best, 450 liked Saint Louis, 1,300 liked Chicago, and the remainder preferred Milwaukee. Develop a frequency table and a relative frequency table to summarize this information.

Step-by-Step Solution

Answer

Milwaukee is preferred by 150 travelers, making up 7.5% of total preferences.

Step 1: Identify Total Responses

Start by identifying the total number of responses provided by the business travelers. We know they received 2,000 responses in total.

Step 2: Count Preference for Each City

List how many travelers prefer each city: 100 liked Indianapolis, 450 liked Saint Louis, and 1,300 liked Chicago. The remaining travelers prefer Milwaukee.

Step 3: Calculate Milwaukee's Preference

To find the number of travelers who prefer Milwaukee, subtract the sum of Indianapolis, Saint Louis, and Chicago preferences from the total responses. \[\text{Milwaukee} = 2000 - (100 + 450 + 1300)\]Thus, 150 travelers prefer Milwaukee.

Step 4: Build Frequency Table

Create a frequency table listing each city and the number of travelers who prefer that city: | City | Frequency | |---------------|-----------| | Indianapolis | 100 | | Saint Louis | 450 | | Chicago | 1300 | | Milwaukee | 150 |

Step 5: Calculate Relative Frequencies

Calculate the relative frequency for each city by dividing the preference frequency by the total number of responses (2000), and then multiply by 100 to convert to a percentage.- Indianapolis: \( \frac{100}{2000} \times 100 = 5\% \)- Saint Louis: \( \frac{450}{2000} \times 100 = 22.5\% \)- Chicago: \( \frac{1300}{2000} \times 100 = 65\% \)- Milwaukee: \( \frac{150}{2000} \times 100 = 7.5\% \)

Step 6: Create Relative Frequency Table

Summarize the information in a relative frequency table: | City | Relative Frequency | |--------------|--------------------| | Indianapolis | 5% | | Saint Louis | 22.5% | | Chicago | 65% | | Milwaukee | 7.5% |

Key Concepts

Understanding Relative Frequency

Relative frequency is a fascinating concept that helps you understand the proportion of specific data compared to the total dataset. In simpler terms, it lets you see what fraction of the whole a particular category represents. Think of it as the percentage of responses or occurrences a certain event gains within a complete set.

It's calculated by taking the frequency (or count) of an event and dividing it by the total number of events. This is then multiplied by 100 to convert it into a percentage. For instance:

• The preference for Indianapolis is calculated as: \( \frac{100}{2000} \times 100 = 5\% \)
• For Saint Louis: \( \frac{450}{2000} \times 100 = 22.5\% \)
• Chicago garners: \( \frac{1300}{2000} \times 100 = 65\% \)
• And Milwaukee gets: \( \frac{150}{2000} \times 100 = 7.5\% \)

Understanding relative frequencies is crucial because it puts data into perspective, telling a more comprehensive story without the overwhelming figures.

Business Statistics and Decision Making

In the world of business statistics, data is gold. Business statistics allows companies to make informed decisions based on clear figures and facts. It involves collecting, analyzing, and interpreting data such as the preferences in the exercise above.

The preference of a city by business travelers is more than just numbers; it's essential for decision-makers in the travel and tourism industry. Knowing that 65% of travelers prefer Chicago can direct marketing campaigns or service improvements towards enhancing experiences in Chicago.

By leveraging these statistics, businesses can:

• Target specific demographics more accurately
• Improve products or services to suit consumer preferences
• Maximize resources and focus efforts on high-return opportunities

Decision-making becomes more data-driven and objective, reducing guesswork and enhancing business strategies aligned with actual consumer behavior.

Data Summarization for Better Insights

Data summarization is key to making information manageable and digestible. It involves condensing raw data into a format that is easy to understand and interpret. In the case of the frequency table provided in the exercise, the data on city preferences is boiled down to a concise summary that shows which cities are favored and by how much.

Summarizing data can take many forms:

• Charts, like bar graphs or pie charts, that visually represent data
• Tables, like our frequency and relative frequency tables, that organize data for quick reference
• Summary statistics such as mean, median, or mode

Using summarization tools reduces complexity and makes it easier to spot trends, patterns, and outliers within the data. It's like creating a snapshot of the overall picture which is invaluable when analyzing large datasets or making strategic, business-related decisions.