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Chapter 15: Problem 30

As described in earlier chapters, many real estate companies and rental agencies now publish their listings on the World Wide Web. One example is the 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 Vacation Rentals, select Beach House 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 next month; indicate that you are willing to spend up to \(\$ 8,000\) per week; and finally click on Search the Cottages. 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.

Short Answer

Expert verified
Use collected data to perform a chi-square test between bathrooms and rental cost; reject null hypothesis if chi-square is high.

Step by step solution

01

Gather and Process Data

Go to the Dunes Realty Company's website and follow the given instructions to search for available beach houses that match the specified criteria: at least 5 bedrooms, 14-person accommodations, oceanfront, no pool or floating dock, available for a period in the next month, with a budget of up to $8,000 per week. Record the data including the number of bathrooms and the weekly rental prices of the listings that match these criteria.
02

Create a Contingency Table

Organize the collected data into a contingency table with two variables: the number of bathrooms and whether the weekly rental price is less than $2,000 or $2,000 or more. Group the listings based on these criteria and count the occurrences for each combination.
03

Combine Categories if Necessary

If any cells in the contingency table have too few observations (typically less than 5), combine them with adjacent cells to ensure all expected values are suitable for the chi-square test. This might involve combining bathroom categories or price categories.
04

Set Up Hypotheses

Formulate the null and alternative hypotheses for the chi-square independence test: \( H_0 \): There is no relationship between the number of bathrooms and the weekly rental cost; \( H_1 \): There is a relationship between the number of bathrooms and the weekly rental cost.
05

Calculate Expected Frequencies

Use the totals from the contingency table to calculate the expected frequency for each cell under the assumption that the variables are independent. Use the formula \( E_{ij} = \frac{(Row \, Total)_i \times (Column \, Total)_j}{ ext{Grand Total}} \).
06

Perform the Chi-Square Test

Calculate the chi-square statistic using the formula \( \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \), where \( O_{ij} \) is the observed frequency in the contingency table and \( E_{ij} \) is the expected frequency. Sum over all cells in the table.
07

Determine Degrees of Freedom

Calculate the degrees of freedom for the chi-square test using the formula \( (r-1)(c-1) \), where \( r \) is the number of rows and \( c \) is the number of columns in the contingency table.
08

Compare Chi-Square Statistic to Critical Value

Look up the critical value of chi-square for the calculated degrees of freedom at the 0.05 significance level using a chi-square distribution table. Compare the calculated chi-square statistic to this critical value to determine whether to reject the null hypothesis.
09

Make a Conclusion

Based on the comparison, decide whether the null hypothesis should be rejected or not. If the chi-square statistic is greater than the critical value, reject \( H_0 \), indicating a significant relationship between the number of bathrooms and the rental cost.

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Key Concepts

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

Contingency Table
A contingency table, often used in statistics, is a type of matrix format that summarizes the relationship between different categorical variables. For our exercise related to real estate listings, the contingency table is used to organize data by two main variables: the number of bathrooms and the weekly rental price category (either less than $2,000 or $2,000 and more). This table helps visualize the data and is crucial before proceeding with any hypothesis testing.

When creating a contingency table, it is important to ensure that all categories have enough data to be meaningful. If some categories have very few observations, as mentioned in the problem, they might need to be combined with adjacent ones. This step is important as it maintains the validity of the statistical tests applied later, like the chi-square test.
Hypothesis Testing
Hypothesis testing is a fundamental statistical method used to decide whether there is enough evidence to reject a null hypothesis. In this exercise, we want to test whether the number of bathrooms is related to the rental cost of beach houses. To do this, we set up two hypotheses:
  • Null Hypothesis (\( H_0 \)): There is no relationship between the number of bathrooms and the weekly rental cost.
  • Alternative Hypothesis (\( H_1 \)): There is a relationship between the number of bathrooms and the weekly rental cost.
This process involves several steps, beginning with setting up the hypotheses, conducting the chi-square test, and finally interpreting the outcomes. A proper setup and interpretation will allow us to make informed decisions based on the observed data.
Degrees of Freedom
Degrees of freedom are an essential concept in statistical testing, prominently featured in the context of a chi-square test. They refer to the number of values in the final calculation of a statistic that are free to vary. In our contingency table analysis, degrees of freedom are calculated as \((r-1)(c-1)\), where \(r\) is the number of rows, and \(c\) is the number of columns in the table.

Understanding degrees of freedom helps in determining the appropriate chi-square distribution and identifying the correct critical value from a chi-square table. The degrees of freedom play a key role in decision-making during hypothesis testing, as they directly affect the test's sensitivity.
Significance Level
The significance level, often denoted by alpha (\( \alpha \)), is a threshold used in hypothesis testing to decide whether a result is statistically significant. In this exercise, the significance level is set at 0.05. This means there is a 5% risk of concluding that a relationship exists between the number of bathrooms and the rental cost when there isn't one.

Setting a 0.05 significance level is a common choice in statistical studies, serving as a balance between being too lenient and too strict. Once the chi-square statistic is calculated, it is compared with the critical value from the chi-square distribution table at the given degrees of freedom and significance level. If the statistic exceeds the critical value, the null hypothesis is rejected, suggesting a significant relationship between the variables.

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Most popular questions from this chapter

A six-sided die is rolled 30 times and the numbers 1 through 6 appear as shown in the following frequency distribution. At the .10 significance level, can we conclude that the die is fair? $$ \begin{array}{|cccc|} \hline \text { Outcome } & \text { Frequency } & \text { Outcome } & \text { Frequency } \\ \hline 1 & 3 & 4 & 3 \\ 2 & 6 & 5 & 9 \\ 3 & 2 & 6 & 7 \\ \hline \end{array} $$

For many years TV executives used the guideline that 30 percent of the audience were watching each of the prime-time networks, that is \(\mathrm{ABC}, \mathrm{NBC}\) and \(\mathrm{CBS},\) and 10 percent were watching cable stations on a weekday night. A random sample of 500 viewers in the Tampa-St. Petersburg, Florida, area last Monday night showed that 165 homes were tuned in to the ABC affiliate, 140 to the CBS affiliate, 125 to the NBC affiliate, and the remainder were viewing a cable station. At the .05 significance level, can we conclude that the guideline is still reasonable?

The following hypotheses are given: \(H_{0}:\) Forty percent of the observations are in category \(A, 40\) percent are in \(B,\) and 20 percent are in \(\mathrm{C}\) \(H_{1}:\) The observations are not as described in \(H_{0}\). We took a sample of \(60,\) with the following results. $$ \begin{array}{|cc|} \hline \text { Category } & f_{o} \\ \hline \mathrm{A} & 30 \\ \mathrm{~B} & 20 \\ \mathrm{C} & 10 \\ \hline \end{array} $$ a. State the decision rule using the .01 significance level. b. Compute the value of chi-square. c. What is your decision regarding \(H_{0} ?\)

Two hundred managers employed in the lumber industry were randomly selected and interviewed regarding their concern about environmental issues. The response of each person was tallied into one of three categories: no concern, some concern, and great concern. The results were: $$ \begin{array}{|lccc|} \hline \begin{array}{l} \text { Level of } \\ \text { Management } \end{array} & \begin{array}{c} \text { No } \\ \text { Concern } \end{array} & \begin{array}{c} \text { Some } \\ \text { Concern } \end{array} & \begin{array}{c} \text { Great } \\ \text { Concern } \end{array} \\ \hline \text { Top management } & 15 & 13 & 12 \\ \text { Middle management } & 20 & 19 & 21 \\ \text { Supervisor } & 7 & 7 & 6 \\ \text { Group leader } & 28 & 21 & 31 \\ \hline \end{array} $$ Use the .01 significance level to determine whether there is a relationship between management level and environmental concern.

In the early 2000 s the Deep Down Mining Company implemented new safety guidelines. Prior to these new guidelines, management expected no accidents in 40 percent of the months, one accident in 30 percent of the months, two accidents in 20 percent of the months, and three accidents in 10 percent of the months. Over the last 10 years, or 120 months, there have been 46 months in which there were no accidents, 40 months in which there was one accident, 22 months in which there were two accidents, and 12 months in which there were 3 accidents. At the .05 significance level can the management at Deep Down conclude that there has been a change in the monthly accident distribution?
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