Chapter 3.2, Problem 36E

a.

To determine

Find the population standard deviation for 2012.

Answer to Problem 36E

• The population standard deviation for 2012 is 96.23.

Explanation of Solution

Calculation:

The price of single family homes in United States for the first quarter of 2012 and 2013 are given.

Standard deviation:

Let $x_1,x_2\mathrm{...}{x}_N$ be the population observations of size N. the population mean is denoted by $\mu$. Hence, the population variance can be expressed as,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{{\displaystyle \sum x^2}-N{\mu }^2}{N}}\end{array}$

Population standard deviation:

Finding mean:

Software procedure:

Step-by-step procedure to find mean using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns 2012.
• Choose option statistics, and select Mean.
• Click OK.

Output using the MINITAB software is given below:

• 
• From the MINITAB output the mean is 201.8.

The deviation, deviation’s squares are calculated below:

x	$\left(x-\mu \right)$	${\left(x-\mu \right)}^2$
87.8	–114	12,996
218.1	16.3	265.69
311.5	109.7	12,034.09
157.2	–44.6	1,989.16
112.5	–89.3	7,974.49
84.9	–116.9	13,665.61
148.2	–53.6	2,872.96
226.4	24.6	605.16
152.1	–49.7	2,470.09
122.1	–79.7	6,352.09
182	–19.8	392.04
147.3	–54.5	2,970.25
363.8	162	26,244
193.5	–8.3	68.89
129.9	–71.9	5,169.61
208.6	6.8	46.24
174.3	–27.5	756.25
103.7	–98.1	9,623.61
359.5	157.7	24,869.29
448	246.2	60,614.44
265.4	63.6	4,044.96
131.9	–69.9	4,886.01
311.6	109.8	12,056.04
Total		${\displaystyle \sum {\left(x-\mu \right)}^2}=21,2967$

Substituting ${\displaystyle \sum {\left(x-\mu \right)}^2}=21,2967$ and N = 23 in the standard deviation formula,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{212,967}{23}}\\ =96.23\end{array}$

• Hence, the population standard deviation for 2012 is 96.23.

b.

To determine

Find the population standard deviation for 2013.

Answer to Problem 36E

• The population standard deviation for 2013 is 115.99.

Explanation of Solution

Calculation:

Population standard deviation:

Finding mean:

Software procedure:

Step-by-step procedure to find mean using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns 2013.
• Choose option statistics, and select Mean.
• Click OK.

Output using the MINITAB software is given below:

• 
• From the MINITAB output the mean is 230.6.

The deviation, deviation’s squares are calculated below:

x	$\left(x-\mu \right)$	${\left(x-\mu \right)}^2$
115.1	–115.5	13,340.25
226.5	–4.1	16.81
332.2	101.6	10,322.56
159.4	–71.2	5,069.44
121	–109.6	12,012.16
101	–129.6	16,796.16
160.4	–70.2	4,928.04
261.2	30.6	936.36
163.7	–66.9	4,475.61
155.1	–75.5	5,700.25
219.9	–10.7	114.49
170.6	–60	3,600
368.2	137.6	18,933.76
197.7	–32.9	1,082.41
169	–61.6	3,794.56
246.5	15.9	252.81
216.7	–13.9	193.21
111	–119.6	14,304.16
412.3	181.7	33,014.89
593.9	363.3	131,986.9
312.6	82	6,724
141.8	–88.8	7,885.44
348.7	118.1	13,947.61
Total		${\displaystyle \sum {\left(x-\mu \right)}^2}=309,431.9$

Substituting ${\displaystyle \sum {\left(x-\mu \right)}^2}=309,431.9$ and N = 23 in the standard deviation formula,

$\begin{array}{c}\sigma =\sqrt{\frac{{\displaystyle \sum {\left(x-\mu \right)}^2}}{N}}\\ =\sqrt{\frac{309,431.9}{23}}\\ =115.99\end{array}$

• Hence, the population standard deviation for 2013 is 115.99.

c.

To determine

Explain whether the spread in house price has increased or not.

Answer to Problem 36E

• The spreads in house price for 2012 has increased in 2013.

Explanation of Solution

• The house price has increased from 2012 to 2013.
• From part (a), the population standard deviation for 2012 is 96.23.
• From part (b), the population standard deviation for 2013 is 115.99.
• The standard deviation for 2012 is less than the population standard deviation of 2013.
• Thus, it can be concluded that the spreads in house price for 2012 has increased in 2013.