Chapter 3.1, Problem 42E

(a)

To determine

To Find: the mean and median price for $2015$ .

Answer to Problem 42E

Price for single family homes in $2015$ , Mean price $\approx 303.3$ and Median price $=270.45$

Explanation of Solution

Given:

  $\begin{array}{l}129.7,151.9,159.5,163.6,175.0,181.5,215.2,217.9,221.1,228.0,231.7,254.5,\\ 286.4,289.0,291.7,314.8,362.9,385.3,398.9,401.1,414.6,449.0,547.8,808.6\end{array}$

Formula Used:

  $\begin{array}{l}\text{Mean}(\overline{x})=\frac{{\displaystyle \sum x_i}}{n}\\ \\ \text{Median}=\left\{\begin{array}{l}\frac{{\left(n+1\right)}^{\text{th }}\text{term}}{2},\text{When}\text{n}\text{is}\text{odd}\\ \frac{{\left(\frac{n}{2}+1\right)}^{\text{th }}\text{term + }{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}}{2},\text{When n is even}\end{array}\right\}\end{array}$

Calculation:

Mean and Median price for single family homes for $2015$ ,

  $\begin{array}{c}\overline{x}=\frac{\sum x_i}{n}\\ =\frac{\left[\begin{array}{l}129.7+151.9+159.5+163.6+175.0+181.5+215.2+217.9+221.1\\ +228.0+231.7+254.5+286.4+289.0+291.7+314.8+362.9\\ +385.3+398.9+401.1+414.6+449.0+547.8+808.6\end{array}\right]}{24}\\ =\frac{7279.1}{24}\\ =303.2958333\approx 303.3\end{array}$

For Median, arrange data in sequential order,

  $\begin{array}{l}129.7,151.9,159.5,163.6,175.0,181.5,215.2,217.9,221.1,228.0,231.7,254.5,\\ 286.4,289.0,291.7,314.8,362.9,385.3,398.9,401.1,414.6,449.0,547.8,808.6\end{array}$

for even numbers

  $\begin{array}{c}\text{Median =}\left(\frac{{\left(\frac{\text{n}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{n}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\frac{\text{24}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{24}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\text{12}\right)}^{\text{th}}\text{term+}{\left(\text{13}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{\text{254}\text{.5+286}\text{.4}}{\text{2}}\right)\\ \text{=}\frac{\text{540}\text{.9}}{\text{2}}\\ \text{=270}\text{.45}\end{array}$

Therefore, price for single family homes, Mean price $\approx 303.3$ and Median price $=270.45$

(b)

To determine

To Find: the mean and median price for $2016$ .

Answer to Problem 42E

Price for single family homes in $2016$ , Mean price $\approx 319.67$ and Median price $=258.3$

Explanation of Solution

Calculation-

Mean and Median price for single family homes for $2016$ ,

  $\begin{array}{c}\overline{x}=\frac{\sum x_i}{n}\\ =\frac{\left[\begin{array}{l}138.1+155.0+160.6+170.3+182.0+199.9+217.4+232.2+232.2+234.7+242.4+250.8+\\ 265.8+310.0+315.5+356.7+394.4+395.5+406.9+420.5+435.8+480.0+589.9+885.6\end{array}\right]}{24}\\ =\frac{7672.1}{24}\\ =319.67\end{array}$

For Median, first arrange data in sequential order, $\begin{array}{l}138.1,155.0,160.6,170.3,182.0,199.9,217.4,232.2,232.2,234.7,242.4,250.8,\\ 265.8,310.0,315.5,356.7,394.4,395.5,406.9,420.5,435.8,480.0,589.9,885.6\end{array}$

  $\begin{array}{c}\text{Median =}\left(\frac{{\left(\frac{\text{n}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{n}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\frac{\text{24}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{24}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\text{12}\right)}^{\text{th}}\text{term+}{\left(\text{13}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{\text{250}\text{.8+265}\text{.8}}{\text{2}}\right)\\ \text{=}\frac{\text{516}\text{.6}}{\text{2}}\\ \text{=258}\text{.3}\end{array}$

Therefore, price for single family homes, Mean price $\approx 319.67$ and Median price $=258.3$

(c)

To determine

To Find: the change in mean price from $2015$ to $2016$ .

Answer to Problem 42E

Mean price increased from $2015$ to $2016$ .

Explanation of Solution

For single family homes in $2015$ , Mean price $\approx 303.3$

For single family homes in $2016$ , Mean price $\approx 319.67$

Mean price increased from $2015$ to $2016$ by approximately $16.37(319.67-303.3=16.37)$

(d)

To determine

To Find: the change in median price from $2015$ to $2016$ .

Answer to Problem 42E

Median price decreased from $2015$ to $2016$ .

Explanation of Solution

For single family homes in $2015$ , Median price $=270.45$

For single family homes in $2016$ , Median price $=258.3$

Median price decreased from $2015$ to $2016$ by $12.15(270.45-258.3=12.15)$