Problem 15 - Basic Statistics for Business and Economics with Student

Question

A real estate agent in the coastal area of Georgia wants to compare the variation in the selling price of homes on the oceanfront with those one to three blocks from the ocean. A sample of 21 oceanfront homes sold within the last year revealed the standard deviation of the selling prices was \(\$ 45,600\). A sample of 18 homes, also sold within the last year, that were one to three blocks from the ocean revealed that the standard deviation was \(\$ 21,330 .\) At the .01 significance level, can we conclude that there is more variation in the selling prices of the oceanfront homes?

Step-by-Step Solution

Answer

Yes, there is more variation in oceanfront home selling prices.

Step 1: State the Hypotheses

To determine if there is more variation in the selling prices of oceanfront homes, we'll conduct an F-test for two variances. The null hypothesis (H0) is that the variances are equal: \( \sigma_1^2 = \sigma_2^2 \). The alternative hypothesis (H1) is that the variance of oceanfront homes is greater: \( \sigma_1^2 > \sigma_2^2 \).

Step 2: Identify the Test Statistic

The test statistic for an F-test is calculated using the formula: \[ F = \frac{s_1^2}{s_2^2} \] where \( s_1 \) is the standard deviation of oceanfront homes and \( s_2 \) is the standard deviation of homes one to three blocks from the ocean. Therefore, \( s_1 = 45600 \) and \( s_2 = 21330 \).

Step 3: Calculate the Test Statistic

Substitute the values into the formula: \[ F = \frac{45600^2}{21330^2} \] First, we square the standard deviations: \( 45600^2 = 2079360000 \) and \( 21330^2 = 455028900 \). Then, calculate \( F \): \[ F = \frac{2079360000}{455028900} \approx 4.57 \]

Step 4: Determine the Critical Value

Using an F-distribution table, find the critical value of F at the \( 0.01 \) significance level with degrees of freedom \( df_1 = 20 \) and \( df_2 = 17 \). The critical value is approximately \( F = 2.578 \).

Step 5: Compare and Make Decision

Compare the calculated F value to the critical value. Since \( 4.57 > 2.578 \), we reject the null hypothesis.

Key Concepts

Understanding Hypothesis Testing

Hypothesis testing is a method used in statistics to make inferences or draw conclusions about a population based on sample data. In the context of our problem, we apply hypothesis testing to compare variances in the selling prices of two groups of homes.
We begin by setting up two hypotheses:

• The null hypothesis (\( H_0 \)) asserts that there is no difference between the variances of the two groups. In mathematical terms, this is stated as: \( \sigma_1^2 = \sigma_2^2 \).
• The alternative hypothesis (\( H_1 \)) suggests that the variance of oceanfront homes (\( \sigma_1^2 \)) is greater than that of homes located one to three blocks away (\( \sigma_2^2 \)).

This forms the basis of the F-test. The conclusion determines whether there is sufficient evidence to support the claim of greater variance in oceanfront homes. If the data significantly supports the alternative hypothesis, we reject the null hypothesis.
Hypothesis testing is essential because it ensures that any conclusions drawn are not just due to random chance, but have a statistical basis.

Exploring Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. In simpler terms, it tells us how spread out the prices are from the average price. In the real estate market context, understanding standard deviation can provide insights into price volatility.
Specifically for our exercise:

• The oceanfront homes have a standard deviation of \( \\( 45,600 \). This larger value suggests a wider range of selling prices, indicating more variation.
• Homes one to three blocks away have a standard deviation of \( \\) 21,330 \), reflecting a narrower price range with less variation.

The standard deviation is central to the F-test because it allows us to compare these data sets in a meaningful way. It's calculated by taking the square root of the variance, which gives us an understanding of how much each data point deviates from the mean. It's a crucial metric for assessing the risk and stability of house prices in different areas.

The Role of Significance Level

In hypothesis testing, the significance level (\( \alpha \)) is a critical concept denoting the probability of rejecting the null hypothesis when it is actually true. It acts as a threshold for determining if an observed effect is statistically significant.
In the current exercise, a significance level of 0.01 is used. This choice implies:

• We accept a 1% risk of mistakenly rejecting the null hypothesis, even if it is true.
• This low level demonstrates a strict criterion, ensuring results are highly reliable.

Choosing the right significance level is important: - Too high a level might lead to false claims in supporting the alternative hypothesis (Type I error).
- Too low a level might miss real effects (Type II error).
In this case, the strict 0.01 level underscores the need for strong evidence before concluding that oceanfront homes experience greater price variation. It assures that the claims made are not just because of random fluctuations in data.