Problem 30 - Basic Statistics for Business and Economics with Student

Question

As described in earlier chapters, many real estate companies and rental agencies now publish their listings on the World Wide Web. One example is Dunes Realty Company located in Garden City, South Carolina, and Surfside Beach, South Carolina. Go to the website http://www.dunes.com and click on Search the Vacation Rentals and then Beach Home Search, then indicate at least 5 bedrooms, accommodations for at least 14 people, oceanfront, and no pool or floating dock; select a period in the current year; indicate that you are willing to spend up to \(\$ 8,000\) per week; and finally click on Search the Beach Homes. Sort the cottages offered into a contingency table by the number of bathrooms and whether the rental price is less than \(\$ 2,000\) for the week or \(\$ 2,000\) or more. You may need to combine some of the cells. Conduct a statistical test to determine if the number of bedrooms is related to the cost. Use the . 05 significance level.

Step-by-Step Solution

Answer

Conduct a Chi-Square test to see if bedrooms and cost are related; reject if p < 0.05.

Step 1: Visit the Website

Navigate to http://www.dunes.com using a web browser. This is where the real estate listings are hosted.

Step 2: Search Vacation Rentals

Click on 'Search the Vacation Rentals' followed by 'Beach Home Search.' This will enable you to input the desired search criteria.

Step 3: Input Search Criteria

Specify at least 5 bedrooms, accommodations for at least 14 people, oceanfront location, and no pool or floating dock. Indicate a budget up to $8,000 per week and select a period within the current year.

Step 4: Conduct the Search

Click on 'Search the Beach Homes' to generate a list of available properties that match your criteria.

Step 5: Sort Results into a Contingency Table

Create a contingency table with one column for properties with a rental price less than $2,000 per week and another for properties priced $2,000 or more. Add a row for each unique number of bathrooms, and fill in the table with the count of properties in each category.

Step 6: Statistical Test for Relationship

Conduct a Chi-Square Test of Independence to determine if there is a statistically significant relationship between the number of bedrooms and the rental price category. Use the .05 significance level.

Step 7: Interpret the Results

If the p-value from the Chi-Square test is less than 0.05, reject the null hypothesis and conclude that the number of bedrooms is related to the cost. Otherwise, do not reject the null hypothesis.

Key Concepts

Contingency Table

A contingency table, sometimes known as a cross-tabulation or crosstab, is a type of table in a matrix format that displays the frequency distribution of variables. In the context of our real estate exercise, the contingency table helps you compare different categories based on their attributes, such as the rental price of beach homes.
Essentially, you create rows and columns to represent categories like the number of bathrooms or rental price brackets. Each cell in the table then shows the count of listings within specific combinations of these categories.
When you are organizing data into a contingency table, follow these basic steps:

• Decide on the factors you want to compare—in this case, number of bathrooms and rental prices.
• Create columns for each rental pricing category (e.g., less than $2,000, $2,000 or more).
• Create rows for the different numbers of bathrooms available.
• Count the number of properties in each category and fill in your table.

This structured approach allows for visual comparison and helps set the foundation for conducting further statistical analysis.

Real Estate Statistics

In real estate statistics, data plays a pivotal role in making informed decisions. Real estate listings capture a variety of details such as prices, number of bedrooms, bathrooms, and location benefits like ocean frontage.
These details are crucial because they directly influence property value and customer choices. Analyzing this data helps in understanding market trends, buyer preferences, and the factors affecting rental pricing.
For example, in the scenario provided, real estate statistics allow us to organize and analyze data to find if there’s a relation between the number of bedrooms and the cost of a rental property.
This involves:

• Gathering data from property listings.
• Organizing that data into a clear format like a contingency table.
• Conducting statistical tests to identify meaningful relationships.

Understanding such statistics provides valuable insights for real estate agents and customers alike, helping them to make better financial and investment decisions.

Significance Level

The significance level, often denoted by \( \alpha \) (alpha), is a critical concept in hypothesis testing. It is the threshold that determines whether or not the results of an experiment or test are statistically significant.
In simpler terms, it’s a way to quantify the level of confidence we have in our test results.

• Commonly used significance levels are 0.05, 0.01, and 0.10.
• For this exercise, a significance level of 0.05 is used. This means there is a 5% chance of rejecting the null hypothesis when it is actually true.

When performing a Chi-Square Test of Independence, the significance level helps determine whether the observed relationship between variables (like bedroom count and rental price) is likely due to chance or if it reflects a true association in the population.
If the p-value obtained from testing is less than or equal to the significance level (\( p \leq \alpha \)), the null hypothesis is rejected, suggesting that the observed data is unlikely under a true null hypothesis. Conversely, if the p-value is greater, we fail to reject the null hypothesis.