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Chapter 13: Problem 5

Suppose that a simple linear regression model is appropriate for describing the relationship between \(y=\) house price (in dollars) and \(x=\) house size (in square feet) for houses in a large city. The population regression line is \(y=23,000+47 x\) and \(\sigma=5000\). a. What is the average change in price associated with one extra square foot of space? With an additional 100 sq. \(\mathrm{ft}\). of space? b. What proportion of 1800 sq. \(\mathrm{ft}\). homes would be priced over \(\$ 110,000\) ? Under \(\$ 100,000\) ?

Short Answer

Expert verified
a. The average price increase per additional square foot is $47 and for an additional 100 sq. ft., the price increase is $4700. b. The proportions of 1800 sq. ft. homes priced over $110,000 and under $100,000 can be found by calculating the z-scores and then finding the area under the standard normal curve using those z-scores.

Step by step solution

01

Calculate the Average Price Increase

The slope of the linear regression model indicates the average change in price associated with one extra square foot of space. Here, the slope is 47, which means the average increase in house price for each extra square foot is $47. For an additional 100 sq. ft. of space, the house price would increase by $47 multiplied by 100, which equals $4700.
02

Determine the Z-Scores

The z-score formula is given by \(Z = \frac{X - \mu}{\sigma}\), where \(X\) represents the house price, \(\mu\) is the mean price, and \(\sigma\) is the standard deviation. First substitute \(X = $110,000\) and \(X = $100,000\) in the given regression equation, and solve each separately to get two equations for the z-scores.
03

Calculate the Proportion of Homes

We then take the z-scores from step 2 and find the area to the right for \(Z_{110000}\) using the standard normal table as \(1 - P(Z <= Z_{110000})\) and to the left for \(Z_{100000}\) as \(P(Z <= Z_{100000})\). The results indicate the proportion of the 1800 sq ft. homes which are priced over $110,000 and under $100,000, respectively.

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Key Concepts

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

House Price Analysis
Analyzing house prices involves understanding how various factors influence the cost of a home. In this case, we're focusing on the relationship between house size and price. This specific example uses a simple linear regression model, represented by the equation \(y = 23,000 + 47x\). This equation suggests two components: a base price of \(23,000 and an increase of \)47 for each additional square foot.

The slope of the equation (47) is key, symbolizing how the house size impacts price. For instance, a home that is 1,800 sq. ft. is expected to cost \(23,000 + 47 \times 1800\), calculated to give the estimated average cost of that house size in the city.

Understanding this linear relationship lets homeowners and potential buyers estimate home values based on size, allowing capacity for future planning or comparison with market trends.
Statistical Modeling
Statistical modeling helps portray and predict relationships between variables. In our house price analysis, simple linear regression is the statistical tool chosen. This choice translates to using one predictor variable (house size) to estimate the dependent variable (house price).

The equation \(y = 23,000 + 47x\) illustrates the linear relationship, where 'x' is the number of square feet, and 'y' is the predicted price. Such a model offers clear insights into predictive analysis. For example, increasing the house size by 100 sq. ft. results in a calculated price increase of $4,700.

Moreover, understanding the standard deviation \(\sigma = 5000\) allows for gauging price variability around the predicted mean. The smaller the standard deviation, the closer the data points are to the prediction line, indicating accuracy and consistency in the model's predictions.
Z-Score Calculation
Z-score calculation is a statistical method to determine how many standard deviations an element is from the mean. This can be useful in assessing the pricing of houses in different scenarios. The formula for a z-score is given by \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the house price, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

For a house priced at \(110,000, we substitute into the regression equation to find the expected mean price, then compute the z-score to compare against the standard normal distribution. This indicates how "above average" this price is when considering the entire market.

Similarly, calculating the z-score for homes priced under \)100,000 provides an understanding of how they rank in terms of affordability. By knowing these scores, stakeholders can easily comprehend the proportion of homes falling above or below certain price thresholds using normal distribution tables.

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Most popular questions from this chapter

Give a brief answer, comment, or explanation for each of the following. a. What is the difference between \(e_{1}, e_{2}, \ldots, e_{n}\) and the \(n\) residuals? b. The simple linear regression model states that \(y=\alpha+\beta x\). c. Does it make sense to test hypotheses about \(b\) ? d. SSResid is always positive. e. A student reported that a data set consisting of \(n=6\) observations yielded residuals \(2,0,5,3,0\), and 1 from the least-squares line. f. A research report included the following summary quantities obtained from a simple linear regression analysis: \(\sum(y-\bar{y})^{2}=615 \quad \sum(y-\hat{y})^{2}=731\)

Reduced visual performance with increasing age has been a much-studied phenomenon in recent years. This decline is due partly to changes in optical properties of the eye itself and partly to neural degeneration throughout the visual system. As one aspect of this problem, the article presented the accompanying data on \(x=\) age and \(y=\) percentage of the cribriform area of the lamina scleralis occupied by pores. \(\begin{array}{llllllllll}x & 22 & 25 & 27 & 39 & 42 & 43 & 44 & 46 & 46 \\ y & 75 & 62 & 50 & 49 & 54 & 49 & 59 & 47 & 54 \\ x & 48 & 50 & 57 & 58 & 63 & 63 & 74 & 74 & \\ y & 52 & 58 & 49 & 52 & 49 & 31 & 42 & 41 & \end{array}\) a. Suppose that prior to this study the researchers had believed that the average decrease in percentage area associated with a 1 -year age increase was \(.5 \%\). Do the data contradict this prior belief? State and test the appropriate hypotheses using a. 10 significance level. b. Estimate true average percentage area covered by pores for all 50 -year- olds in the population in a way that conveys information about the precision of estimation.

The article reported the following data on maximum outdoor temperature \((x)\) and hours of chiller operation per day \((y)\) for a 3 -ton residential gas air- conditioning system: $$ \begin{array}{rrrrrrr} x & 72 & 78 & 80 & 86 & 88 & 92 \\ y & 4.8 & 7.2 & 9.5 & 14.5 & 15.7 & 17.9 \end{array} $$ Suppose that the system is actually a prototype model, and the manufacturer does not wish to produce this model unless the data strongly indicate that when maximum outdoor temperature is \(82^{\circ} \mathrm{F}\), the true average number of hours of chiller operation is less than \(12 .\) The appropriate hypotheses are then $$ H_{0}: \alpha+\beta(82)=12 \text { versus } H_{a}: \alpha+\beta(82)<12 $$ Use the statistic $$ t=\frac{a+b(82)-12}{s_{a+b(82)}} $$ which has a \(t\) distribution based on \((n-2)\) df when \(H_{0}\) is true, to test the hypotheses at significance level \(.01\).

A random sample of \(n=347\) students was selected, and each one was asked to complete several questionnaires, from which a Coping Humor Scale value \(x\) and a Depression Scale value \(y\) were determined. The resulting value of the sample correlation coefficient was \(-.18\). a. The investigators reported that \(P\) -value \(<.05 .\) Do you agree? b. Is the sign of \(r\) consistent with your intuition? Explain. (Higher scale values correspond to more developed sense of humor and greater extent of depression.) c. Would the simple linear regression model give accurate predictions? Why or why not?

If the sample correlation coefficient is equal to 1, is it necessarily true that \(\rho=1\) ? If \(\rho=1\), is it necessarily true that \(r=1 ?\)
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