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Chapter 11: Problem 5

What did housing prices look like in the "good old days"? The median sale prices for new single family houses are given in the accompanying table for the years 1972 through \(1979 .^{\star}\) Letting \(Y\) denote the median sales price and \(x\) the year (using integers \(1,2, \ldots, 8\) ), fit the model \(Y=\beta_{0}+\beta_{1} x+\varepsilon .\) What can you conclude from the results?

Short Answer

Expert verified
Housing prices increased over these years, as shown by the positive slope.

Step by step solution

01

Organize the Data

First, we need to list the data from the table, where each year from 1972 to 1979 corresponds to an integer value from 1 to 8. This allows us to have pairs of data (x, Y) for each year and its corresponding median sales price.
02

Formulate the Linear Model

The linear regression model is given by \( Y = \beta_0 + \beta_1 x + \varepsilon \), where \( Y \) is the median sales price, \( x \) is the year (represented by integer values from 1 to 8), \( \beta_0 \) is the y-intercept, \( \beta_1 \) is the slope of the line (price change per year), and \( \varepsilon \) is the error term.
03

Calculate Required Statistics

Next, calculate sums such as \( \sum x \), \( \sum Y \), \( \sum x^2 \), and \( \sum xY \). These values are necessary for applying the formulas to calculate the slope \( \beta_1 \) and y-intercept \( \beta_0 \).
04

Calculate Slope (\(\beta_1\))

Use the formula \( \beta_1 = \frac{N(\sum xY) - (\sum x)(\sum Y)}{N(\sum x^2) - (\sum x)^2} \) to calculate the slope, where \( N \) is the number of data points (in this case, 8).
05

Calculate Y-Intercept (\(\beta_0\))

Calculate \( \beta_0 \) using the formula \( \beta_0 = \frac{(\sum Y)(\sum x^2) - (\sum x)(\sum xY)}{N(\sum x^2) - (\sum x)^2} \).
06

Construct the Regression Equation

Substitute the calculated values of \( \beta_0 \) and \( \beta_1 \) into the model equation \( Y = \beta_0 + \beta_1 x \). This provides the specific relationship for median sales prices over the given years.
07

Conclusion from the Model

Evaluate the slope \( \beta_1 \) from the regression equation. A positive \( \beta_1 \) would indicate that housing prices increased over the years, while a negative \( \beta_1 \) would suggest they decreased. The strength of \( \beta_1 \) gives insights into the rate of change.

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Key Concepts

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

statistical modeling
Statistical modeling is a powerful tool used in various fields to understand and predict patterns through mathematical formulas. At its core, statistical modeling involves creating equations that describe relationships in data. Linear regression is one such model that helps to establish a linear relationship between a dependent and an independent variable. Here, the dependent variable is often affected by various other factors and the goal is to see how one isolated variable influences it specifically.
In the context of the housing market problem, statistical modeling is used to explore how median housing prices (the dependent variable) changed over several years (the independent variable). This relationship can be represented using the formula \( Y = \beta_0 + \beta_1 x + \varepsilon \) where \( Y \) is the dependent variable representing the median sale price, \( \beta_0 \) is the starting point of the prices (y-intercept), \( \beta_1 \) represents the change in price per year (slope), and \( \varepsilon \) accounts for any error or variance not captured by the model.
Ultimately, statistical models like linear regression provide a simplified way to make sense of historical data and have potential future applications. By fitting a model to past data, we can extrapolate trends that might suggest future outcomes. This, in turn, helps stakeholders make informed decisions.
median sales price analysis
Median sales price analysis is crucial in understanding the central trend of a dataset. The median value is often preferred over the mean because it is not affected by extreme values or outliers, giving a more accurate picture of the market. When analyzing the housing market, the median sales price reveals what a typical home might have sold for during a specified time period.
To analyze the median sales price using linear regression, one would first gather data over several years, assigning an integer to each year for easy data handling. This creates pairs of the year and the median sales price for analysis. Then, using the methods provided in the solution, one calculates the regression coefficients \( \beta_0 \) and \( \beta_1 \). This allows for the construction of a regression equation to understand price trends over time.
Analyzing these trends can provide insights into patterns that may exist in the data:
  • A positive slope \( \beta_1 \) implies an upward trend, indicating that housing prices tend to increase overtime.
  • A negative slope would suggest a downward trend in pricing.
This type of analysis is indispensable for stakeholders such as policymakers, real estate investors, and potential buyers.
housing market trends
Understanding housing market trends is key for anyone interested in real estate, whether they are buying, selling, or investing. These trends can reflect broader economic conditions and changes in supply and demand dynamics. Trends over time allow us to predict future price movements and adapt our strategies accordingly.
By examining the historical data on median sales prices, we can identify how the market has evolved. In the exercise scenario, linear regression results in an equation where the slope \( \beta_1 \) reveals overall market direction. If this slope is positive, it indicates that the market has seen consistent growth through those years, suggesting a trend towards increasing property values.
Housing market trends are often influenced by several factors including:
  • Interest rates
  • Economic growth
  • Government policies
  • Demographic changes
The linear regression model helps boil down these complex elements into understandable insights. Overall, careful analysis of housing market trends can empower better decision-making, help mitigate risks, and maximize potential revenue for homes being bought, sold, or rented.

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Most popular questions from this chapter

If \(\widehat{\beta}_{0}\) and \(\widehat{\beta}_{1}\) are the least-squares estimates for the intercept and slope in a simple linear regression model, show that the least-squares equation \(\hat{y}=\hat{\beta}_{0}+\hat{\beta}_{1} x\) always goes through the point \((\bar{x}, \bar{y})\). [Hint: Substitute \(\bar{x}\) for \(x\) in the least-squares equation and use the fact that \(\left.\widehat{\beta}_{0}=\bar{y}-\widehat{\beta}_{1} \bar{x} .\right]\)

Information about eight four-cylinder automobiles judged to be among the most fuel efficient in 2006 is given in the following table. Engine sizes are in total cylinder volume, measured in liters (L). $$\begin{array}{lcc}\text { Car } & \text { Cylinder Volume }(x) & \text { Horsepower }(y) \\\\\hline \text { Honda Civic } & 1.8 & 51 \\ \text { Toyota Prius } & 1.5 & 51 \\\\\text { WW Golf } & 2.0 & 115 \\\\\text { WW Beetle } & 2.5 & 150 \\\\\text { Toyota Corolla } & 1.8 & 126 \\ \text { WW Jetta } & 2.5 & 150 \\\\\text { Mini Cooper } & 1.6 & 118 \\\\\text { Toyota Yaris } & 1.5 & 106\end{array}$$ a. Plot the data points on graph paper. b. Find the least-squares line for the data. c. Graph the least-squares line to see how well it fits the data. d. Use the least-squares line to estimate the mean horsepower rating for a fuel-efficient automobile with cylinder volume \(1.9 \mathrm{L}\).

Consider the general linear model $$Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\cdots+\beta_{k} x_{k}+\varepsilon$$ where \(E(\varepsilon)=0\) and \(V(\varepsilon)=\sigma^{2} .\) Notice that \(\widehat{\beta}_{i}=\mathbf{a}^{\prime} \hat{\boldsymbol{\beta}},\) where the vector a is defined by $$a_{j}=\left\\{\begin{array}{ll} 1, & \text { if } j=i, \\ 0, & \text { if } \neq i \end{array}\right.$$ Use this to verify that \(E\left(\widehat{\beta}_{i}\right)=\beta_{i}\) and \(V\left(\widehat{\beta}_{i}\right)=c_{i i} \sigma^{2},\) where \(c_{i i}\) is the element in row \(i\) and column \(i\) of \(\left(\mathbf{x}^{\prime} \mathbf{x}\right)^{-1}\)

The accompanying table gives the catches of Peruvian anchovies (in millions of metric tons) and the prices of fish meal (in current dollars per ton) for 14 consecutive years. $$\begin{array}{|c|cccccccccccccc|}\hline \text { Price of fish meal }(y) & 190 & 160 & 134 & 129 & 172 & 197 & 167 & 239 & 542 & 372 & 245 & 376 & 454 & 410 \\\\\hline \text { Anchovy catch }(x) & 7.23 & 8.53 & 9.82 & 10.26 & 8.96 & 12.27 & 10.28 & 4.45 & 1.78 & 4.0 & 3.3 & 4.3 & 0.8 & 0.5 \\\\\hline\end{array}$$ a. Find the least-squares line appropriate for these data. b. Plot the points and graph the line as a check on your calculations.

In the biological and physical sciences, a common model for proportional growth over time is $$ E(Y)=1-e^{-\beta t} $$ where \(Y\) denotes a proportion and \(t\) denotes time. \(Y\) might represent the proportion of eggs that hatch, the proportion of an organism filled with diseased cells, the proportion of patients reacting to a drug, or the proportion of a liquid that has passed through a porous medium. With \(n\) observations of the form \(\left(y_{i}, t_{i}\right),\) outline how you would estimate and then form a confidence interval for \(\beta\)
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