Chapter 4, Problem 1PE

Data obtained from a county auditor in the Excel file Home Market Value provide information about the age, square footage, and current market value of houses along one street in a particular subdivision. Considering these data as a population of homeowners on this street, compute the mean, variance, and standard deviation for each of these variables using a spreadsheet and formulas (4.1), (4.4), and (4.6). Verify your calculations using the appropriate Excel function.

To determine

To find population mean, variance and standard deviation and verify it using excel function for each of the variables.

Explanation of Solution

Given:

Data,

House Age	Square Feet	Market Value
33	1,812	$90,000.00
32	1,914	$104,400.00
32	1,842	$93,300.00
33	1,812	$91,000.00
32	1,836	$101,900.00
33	2,028	$108,500.00
32	1,732	$87,600.00
33	1,850	$96,000.00
32	1,791	$89,200.00
33	1,666	$88,400.00
32	1,852	$100,800.00
32	1,620	$96,700.00
32	1,692	$87,500.00
32	2,372	$114,000.00
32	2,372	$113,200.00
33	1,666	$87,500.00
32	2,123	$116,100.00
32	1,620	$94,700.00
32	1,731	$86,400.00
32	1,666	$87,100.00
28	1,520	$83,400.00
27	1,484	$79,800.00
28	1,588	$81,500.00
28	1,598	$87,100.00
28	1,484	$82,600.00
28	1,484	$78,800.00
28	1,520	$87,600.00
27	1,701	$94,200.00
28	1,484	$82,000.00
28	1,468	$88,100.00
28	1,520	$88,100.00
27	1,520	$88,600.00
27	1,484	$76,600.00
28	1,520	$84,400.00
27	1,668	$90,900.00
28	1,588	$81,000.00
28	1,784	$91,300.00
27	1,484	$81,300.00
27	1,520	$100,700.00
28	1,520	$87,200.00
27	1,684	$96,700.00
27	1,581	$120,700.00

Formula:

Population mean:

  $\mu =\frac{{\displaystyle \sum _{i=1}^N{X}_i}}{N}$

Population variance:

  ${\sigma }^2=\frac{{\displaystyle \sum {(x_i-\overline{x})}^2}}{N}$

Population standard deviation:

  $\sigma =\sqrt{\sigma }$

Calculation:

For each of the variable, need to find population mean, variance and standard deviation.

Variable House Age:

Total number of values = N=42

Mean = $\mu =\frac{1253}{42}=29.83$

Using excel function: =AVERAGE(A2:A43)

Mean = 29.83 which is exactly matched with calculated mean.

Population variance:

  ${\sigma }^2=\frac{1}{42}\times \frac{1451}{6}=5.76$

Population standard deviation:

  $\sigma =\sqrt{{\sigma }^2}=\sqrt{5.76}=2.40$

Using excel formula:

=VAR.P(A2:A43)

Variance = 5.76

=STDEV.P(A2:A43)

Standard deviation = 2.40

Both the values almost match with calculated value.

Variable square feet:

Mean = $\mu =\frac{71201}{42}=1695.262$

Using excel: =AVERAGE(B2:B43)

Mean = 1695.26 which is almost match with calculated value.

Population variance:

  ${\sigma }^2=47357.96$

Population standard deviation:

  $\sigma =\sqrt{{\sigma }^2}=\sqrt{47357.96}=217.62$

Using excel:

Population variance: =VAR.P(B2:B43) which gives 47357.96

Population standard deviation found by excel is matched with calculated variance.

Standard deviation: =STDEV.P(B2:B43) which gives 217.62

Standard deviation is also matched with calculated value.

Variable Market value:

Mean : $\mu =92069.05$

Using excel, =AVERAGE(C2:C43) which gives 92069.05

Population variance:

  ${\sigma }^2=108715946.712$

Population standard deviation:

  $\sigma =\sqrt{{\sigma }^2}=\sqrt{108715946.712}=10426.69$

Using excel:

Population variance: =VAR.P(C2:C43) which gives 108715946.71

Population standard deviation: =STDEV.P(C2:C43) which gives 10426.69

Values found by excel and calculated values are same.