Chapter 3.2, Problem 40E

House prices: The following table presents prices, in thousands of dollars, of single-family homes for some of the largest metropolitan areas in the United States for the third quarter of 2015 and the third quarter of 20l6.

1. Find the population standard deviation for 2015.
2. Find the population standard deviation for 2016.
3. In general, house prices increased from 2015 to 2016. Did the spread in house prices increase as well, or did it decrease?

To determine

To find the Population standard deviation for 2015.

Answer to Problem 40E

Standard Deviation = 149.71

Explanation of Solution

Given:

The data

Metro Area	2015	2016	Metro Area	2015	2016
Atlanta, GA	181.5	182	New York, NY	398.3	395.4
Baltimore, MD	254.5	265.8	Philadelphia, PA	231.7	232.2
Boston, MA	414.6	435.8	Phoenix, AZ	217.9	234.7
Chicago, IL	286.4	250.8	Pittsburgh, PA	159.5	155
Cincinnati, OH	151.9	160.6	Portland, OR	314.8	356.7
Cleveland, OH	129.7	138.1	Riverside, CA	291.7	315.5
Dallas, TX	215.2	232.2	St. Louis, MO	163.6	170.3
Denver, CO	362.9	394.4	San Diego, CA	547.8	589.9
Houston, TX	221.1	217.4	San Francisco, CA	808.6	885.6
Los Angeles, CA	449	480	Seattle, WA	385.3	420.5
Miami, FL	289	310	Tampa, FL	175	199.9
Minneapolis, MN	228	242.4	Washington, DC	401.1	406.9

Calculation:

Metro Area	2015 (x)	${\left(x_i-\overline{x}\right)}^2$
Atlanta, GA	181.5	14834.23
Baltimore, MD	254.5	2381.03
Boston, MA	414.6	12388.62
Chicago, IL	286.4	285.47
Cincinnati, OH	151.9	22920.70
Cleveland, OH	129.7	30135.51
Dallas, TX	215.2	7760.88
Denver, CO	362.9	3552.66
Houston, TX	221.1	6756.16
Los Angeles, CA	449	21229.70
Miami, FL	289	204.37
Minneapolis, MN	228	5669.46
New York, NY	398.3	9025.79
Philadelphia, PA	231.7	5125.96
Phoenix, AZ	217.9	7292.45
Pittsburgh, PA	159.5	20677.24
Portland, OR	314.8	132.35
Riverside, CA	291.7	134.46
St. Louis, MO	163.6	19514.93
San Diego, CA	547.8	59782.29
San Francisco, CA	808.6	255332.30
Seattle, WA	385.3	6724.68
Tampa, FL	175	16459.82
Washington, DC	401.1	9565.66
SUM	7279.1	537886.71
Mean	303.296
Standard deviation	149.71

  $\begin{array}{c}\text{Mean = }\frac{\sum x_i}{n}\\ =\frac{7279.1}{24}\\ =303.296\end{array}$

  $\begin{array}{c}\text{Standard Deviation = }\sqrt{\frac{\sum {\left(x_i-\overline{x}\right)}^2}{n}}\\ =\sqrt{\frac{537886.71}{24}}\\ =149.71\end{array}$

The mean and standard deviation for2015 is 303.296 and 149.71.

To determine

To find the Population standard deviation for 2016.

Answer to Problem 40E

Standard Deviation = 164.65

Explanation of Solution

Calculation:

Metro Area	2016	${\left(x_i-\overline{x}\right)}^2$
Atlanta, GA	182	18953.26
Baltimore, MD	265.8	2902.07
Boston, MA	435.8	13485.98
Chicago, IL	250.8	4743.19
Cincinnati, OH	160.6	25303.53
Cleveland, OH	138.1	32967.97
Dallas, TX	232.2	7651.15
Denver, CO	394.4	5584.45
Houston, TX	217.4	10459.32
Los Angeles, CA	480	25705.44
Miami, FL	310	93.53
Minneapolis, MN	242.4	5970.78
New York, NY	395.4	5734.91
Philadelphia, PA	232.2	7651.15
Phoenix, AZ	234.7	7220.04
Pittsburgh, PA	155	27116.48
Portland, OR	356.7	1371.16
Riverside, CA	315.5	17.40
St. Louis, MO	170.3	22311.65
San Diego, CA	589.9	73023.80
San Francisco, CA	885.6	320275.82
Seattle, WA	420.5	10166.52
Tampa, FL	199.9	14345.05
Washington, DC	406.9	7608.93
SUM	7672.1	650663.57
Mean	319.67
Standard deviation	164.65

  $\begin{array}{c}\text{Mean = }\frac{\sum x_i}{n}\\ =\frac{7672.1}{24}\\ =319.67\end{array}$

  $\begin{array}{c}\text{Standard Deviation = }\sqrt{\frac{\sum {\left(x_i-\overline{x}\right)}^2}{n}}\\ =\sqrt{\frac{650663.57}{24}}\\ =164.65\end{array}$

The mean and standard deviation of sports model is 319.67 and 164.65.

To determine

To explain whether the house prices has increased from 2015 to 2016 or not and also explain the spread from 2015 to 2016.

Answer to Problem 40E

Yes, House price and spread has increasedfrom 2015 to 2016.

Explanation of Solution

Mean house price in 2015 was 303.296 and in 2016 it was 319.67 which shows that house prices has increased from 2015 to 2016 and also the spread has increased from 149.71 to 164.65.