91Ó°ÊÓ

Multiple choice: Select the best answer for Exercises, which are based on the following information. To determine property taxes, Florida reappraises real estate every year, and the county appraiser's Web site lists the current "fair market value" of each piece of property. Property usually sells for somewhat more than the appraised market value. We collected data on the appraised market values \(x\) and actual selling prices \(y\) (in thousands of dollars) of a random sample of 16 condominium units in Florida. We checked that the conditions for inference about the slope of the population regression line are met. Here is part of the Minitab output from a least-squares regression analysis using these data. \({ }^{13}\) $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 127.27 & 79.49 & 1.60 & 0.132 \\ \text { Appraisal } & 1.0466 & 0.1126 & 9.29 & 0.000 \\ \mathrm{~S}=69.7299 & \mathrm{R}-\mathrm{Sq}=86.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =85.1 \% \end{array} $$ Is there convincing evidence that selling price increases as appraised value increases? To answer this question, test the hypotheses (a) \(\quad H_{0}: \beta=0\) versus \(H_{a}: \beta > 0\). (b) \(H_{0}: \beta=0\) versus \(H_{a}: \beta < 0\). (c) \(\quad H_{0}: \beta=0\) versus \(H_{a}: \beta \neq 0\). (d) \(H_{0}: \beta > 0\) versus \(H_{a}: \beta=0\). (e) \(\quad H_{0}: \beta=1\) versus \(H_{a}: \beta>1\)

Step by step solution

01

Identify the Regression Equation Coefficient

The coefficient of the 'Appraisal' in the regression output is 1.0466. This represents the estimated change in the selling price (in thousands of dollars) for each 1 unit (thousand dollars) increase in the appraised value.

02

Determine the Null and Alternative Hypotheses

We need to test if there is a relationship between the appraised value and the selling price. This corresponds to testing if the slope \( \beta eq 0 \). Thus, the appropriate hypotheses are \( H_0: \beta = 0 \) and \( H_a: \beta > 0 \).

03

Analyze the P-value

The P-value for the 'Appraisal' coefficient is 0.000. A P-value of 0.000 (significantly lower than 0.05) indicates strong evidence against \( H_0 \), so we reject \( H_0 \) in favor of \( H_a \).

04

Evaluate the Hypotheses

The evidence suggests that there is a strong positive relationship between the appraised value and the selling price, implying that the selling price increases as the appraised value increases. Hence, the correct choice aligns with the hypothesis test \( (a) \).

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Key Concepts

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

Hypothesis Testing

In statistical analysis, hypothesis testing is a method used to determine whether there is enough evidence in a sample of data to infer a condition for the entire population. It involves the formulation of two mutually exclusive statements, known as the null hypothesis ( H_0 ) and the alternative hypothesis ( H_a ). The null hypothesis is the default assumption that there is no effect or no difference, while the alternative hypothesis is what you aim to evidence.

In the context of regression analysis, hypothesis testing is often used to determine if there is a meaningful relationship between variables. Specifically, we test the slope ( β ) of the regression line. For the given problem relating to property values, the null hypothesis ( H_0: β = 0 ) suggests that there is no relationship between the appraised value and selling price, while the alternative hypothesis ( H_a: β > 0 ) suggests that as the appraised value increases, the selling price is expected to increase as well.

• Null hypothesis ( H_0 ): No change in selling price with changes in appraised value.
• Alternative hypothesis ( H_a ): Selling price increases with appraised value.

Linear Regression

Linear regression is a statistical method that models the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. In simple linear regression, we deal with two variables, where one is an independent variable (x) and the other is a dependent variable (y).

The regression line is represented by the equation \( y = β_0 + β_1x \), where \( β_0 \) is the y-intercept, and \( β_1 \) is the slope. The slope informs us about the change in the dependent variable for a one-unit change in the independent variable.

In this exercise, the appraised values are the independent variable (x), and the actual selling prices are the dependent variable (y). The coefficient of the appraisal, 1.0466, indicates that for every \(1,000 increase in appraised value, the selling price increases by about \)1,046.60 on average.

P-value Analysis

P-value analysis helps assess the strength of evidence against the null hypothesis. It is the probability of observing the test results, assuming the null hypothesis is true.

In hypothesis testing, a low P-value signifies strong evidence against the null hypothesis. Conventionally, a P-value threshold of 0.05 is used. A P-value less than 0.05 suggests that what we have observed is statistically significant, warranting a rejection of the null hypothesis.

In the Minitab output for the real estate regression, the P-value for the appraisal coefficient is reported as 0.000. This very low P-value strongly suggests that the relationship between appraised value and selling price is significant, leading to the rejection of the null hypothesis in favor of the alternative hypothesis.

Slope Interpretation

Understanding the slope in regression analysis is crucial for interpreting the data relationships. The slope ( β_1 ) in the regression equation represents the average change in the dependent variable for each one-unit change in the independent variable.

In this particular exercise, the slope of the regression line is 1.0466. This means that for every $1,000 increase in the appraised value of the property, the selling price is expected to increase by approximately $1,046.60 on average. A positive slope like this indicates a direct or positive relationship between the two variables.

• A positive slope suggests that the dependent variable increases as the independent variable increases.
• If the slope was negative, it would suggest an inverse relationship.