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Statistics / Statistics: Informed Decisions Using Data 4 / Chapter 4 / Problem 12RE

Statistics: Informed Decisions Using Data | 4th Edition | ISBN: 9780321757272 | Authors: Michael Sullivan, III

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Apartment Rents Using the Nassau County apartment rent

Chapter 4, Problem 12RE

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QUESTION:

Problem 12RE

Apartment Rents Using the Nassau County apartment rent data and the results of Problem 3, do the following:

(a) Find the least-squares regression line treating square footage as the explanatory variable.

(b) Are there any outliers or influential observations?

Questions & Answers

QUESTION:

Problem 12RE

Apartment Rents Using the Nassau County apartment rent data and the results of Problem 3, do the following:

(a) Find the least-squares regression line treating square footage as the explanatory variable.

(b) Are there any outliers or influential observations?

ANSWER:

Answer

Step 1 of 2

(a)

Using the Nassau County apartment rent data, we need to find the following.

We are asked to find the least-squares regression line treating square footage as the explanatory variable.

Square Footage, $x$	Rent per Month, $y$
1100	1875
588	1075
1250	1775
556	1050
825	1300
743	1475
660	1315
975	1400
1429	1900
800	1650
1906	4625
1077	1395

Sample Linear Correlation Coefficient can be found by the following formula,

${r}_{xy}=\frac{\sum\limits_{\ }^{\ }\left({\frac{{x}_{i}-\overline{x}}{{S}_{x}}}\right)\left({\frac{{y}_{i}-\overline{y}}{{S}_{y}}}\right)}{n-1}$

…..(1)

Square Footage, $x$	Rent per Month, $y$
1100	1875
588	1075
1250	1775
556	1050
825	1300
743	1475
660	1315
975	1400
1429	1900
800	1650
1906	4625
1077	1395
$\overline{x}=992.42$	$\overline{y}=1736.25$
${S}_{x}=393.4$	${S}_{y}=952.2035$

Substitute above values into the equation (1), we get,

${r}_{xy}=\frac{\sum\limits_{\ }^{\ }\left({\frac{{x}_{i}-\overline{x}}{{S}_{x}}}\right)\left({\frac{{y}_{i}-\overline{y}}{{S}_{y}}}\right)}{n-1}=\ 0.8674$

The equation of the least-squares regression line is given by

$\widehat{y}={b}_{1}x+{b}_{0}$

Where

${b}_{1}=r.\frac{{S}_{y}}{{S}_{x}}$ is the slope of the least-squares regression line.

And

${b}_{0}=\overline{y}-{b}_{1}\overline{x}$ is the y-intercept of the least-squares regression line

x is the sample mean and ${S}_{x}$ is the sample standard deviation of the explanatory variable x;

y is the sample mean and ${S}_{y}$ is the sample standard deviation of the response variable y.

${b}_{1}=r.\frac{{S}_{y}}{{S}_{x}}=\frac{\ 0.8674\times{}952.2035}{393.4}=2.1$

${b}_{0}=1736.25-(2.1)992.42=\ -347.35$

$\widehat{y}\ =2.1x\ -347.35$

Hence the least-squares regression line treating square footage as the explanatory variable is $\widehat{y}\ =2.1x\ -347.35.$

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