Chapter 2, Problem 51DA

Refer to the North Valley Real Estate data that reports information on homes sold during the last year. For the variable price, select an appropriate class interval and organize the selling prices into a frequency distribution. Write a brief report summarizing your findings. Be sure to answer the following questions in your report.

1. a. Around what values of price do the data tend to cluster?
2. b. Based on the frequency distribution, what is the typical selling price in the first class? What is the typical selling price in the last class?
3. c. Draw a cumulative relative frequency distribution. Using this distribution, fifty percent of the homes sold for what price or less? Estimate the lower price of the top ten percent of homes sold. About what percent of the homes sold for less than $300,000?
4. d. Refer to the variable bedrooms. Draw a bar chart showing the number of homes sold with 2, 3, 4 or more bedrooms. Write a description of the distribution.

To determine

Find an appropriate class interval.

Create a frequency distribution for the selling prices and explain the results.

At what values of price the data tend to cluster.

Answer to Problem 51DA

The frequency distribution for the selling price is given below:

Selling price (1,000’s)	Frequency	Cumulative frequency
120-240	26	26
240-360	36	$26+36=62$
360-480	27	$62+27=89$
480-600	7	$89+7=96$
600-720	4	$96+4=100$
720-840	2	$100+2=102$
840-960	3	$102+3=105$
Total	105

Explanation of Solution

Selection of number of classes:

“2 to the k rule” suggests that the number of classes is the smallest value of k, where $2^k$ is greater than the number of observations.

It is given that the data set consists of 105 observations. The value of k can be obtained as follows:

$\begin{array}{l}{2}^6=64<105\\ 2^7=128>105\end{array}$

Here, $k=7$ is the smallest value for which $2^k$ is greater than the number of observations.

Therefore, the number of classes for the given data set is 7.

From the data set North Valley Real Estate Data, the maximum and minimum values are 919,480 and 167,962, respectively.

The formula for the class interval is given as follows:

$i\ge \frac{\left(\text{Maximum value}-\text{Minimum value}\right)}{k}$

Where, i is the class interval and k is the number of classes.

Therefore, the class interval for the given data can be obtained as follows:

$\begin{array}{c}i\ge \frac{\left(\text{Maximum value}-\text{Minimum value}\right)}{k}\\ \ge \frac{919,480-167,962}{7}\\ \ge \frac{751,518}{7}\\ \ge 107,359.7\end{array}$

In practice, the class interval size is usually rounded up to some convenient number. Therefore, the reasonable class interval is 120,000.

Frequency distribution:

The frequency table is a collection of mutually exclusive and exhaustive classes, which shows the number of observations in each class.

Since the minimum value is 167,962 and the class interval is 120,000, the first class would be 120,000-240,000 or 160,000-280,000. Here, the first one is preferred as the first class of the frequency distribution. The frequency distribution for the selling price can be constructed as follows:

Selling price (1,000’s)	Frequency	Cumulative frequency
120-240	26	26
240-360	36	$26+36=62$
360-480	27	$62+27=89$
480-600	7	$89+7=96$
600-720	4	$96+4=100$
720-840	2	$100+2=102$
840-960	3	$102+3=105$
Total	105

From the above table, 89 out of 105 homes are sold between $120,000 and $480,000.

To determine

Find the typical selling in first class.

Find the typical selling in last class.

Answer to Problem 51DA

The typical selling price of first class is $180,000.

The typical selling price of last class is $900,000.

Explanation of Solution

The lower and upper limits of the first class are $120,000 and $240,000.

The typical selling price of first class is calculated as follows:

$\begin{array}{c}\text{Typical selling price}=\frac{\text{Lower limit + Upper limit}}{2}\\ =\frac{120,000+240,000}{2}\\ =\$180,000\end{array}$

Thus, typical selling price of first class is $180,000.

The lower and upper limits of the last class are $840,000 and $960,000.

The typical selling price of last class is calculated as follows:

$\begin{array}{c}\text{Typical selling price}=\frac{\text{Lower limit + Upper limit}}{2}\\ =\frac{840,000+960,000}{2}\\ =\$900,000\end{array}$

Thus, typical selling price of last class is $900,000.

To determine

Create a cumulative relative frequency polygon for the frequency distribution.

Identify the price for which less than 50% of the homes are sold.

Find the lower price of the top 10% of homes are sold.

About what percentage of homes are sold for less than $300,000.

Answer to Problem 51DA

The cumulative frequency polygon for the given data is as follows:

There are 50% of the homes sold approximately less than $254,000.

The top 10% of homes are sold for at least $520,000.

There are 59% of homes sold for less than $300,000.

Explanation of Solution

For the given data set, the cumulative relative frequency table with midpoints of classes is obtained as follows:

Selling price (1,000’s)	Midpoint	Cumulative frequency	Relative cumulative frequency
120-240	$\frac{120+240}{2}=180$	26	$\frac{26}{105}=0.248$
240-360	$\frac{240+360}{2}=300$	62	$\frac{62}{105}=0.590$
360-480	$\frac{360+480}{2}=420$	89	$\frac{89}{105}=0.848$
480-600	$\frac{480+600}{2}=540$	96	$\frac{96}{105}=0.914$
600-720	$\frac{600+720}{2}=660$	100	$\frac{100}{105}=0.952$
720-840	$\frac{720+840}{2}=780$	102	$\frac{102}{105}=0.971$
840-960	$\frac{840+960}{2}=900$	105	$\frac{105}{105}=1$

The cumulative relative frequency polygon for the given data can be drawn using EXCEL.

Step-by-step procedure to obtain the frequency polygon using EXCEL is as follows:

• Enter the column of midpoints along with the cumulative relative frequency column.
• Select the total data range with labels.
• Go to Insert > Charts > line chart.
• Select the appropriate line chart.
• Click OK.

From the above cumulative relative frequency polygon, 50% of the homes are sold approximately less than $254,000.

The top 10% of homes are sold for at least $520,000.

There are 59% of homes sold for less than $300,000.

To determine

Create a bar chart for the number of bedrooms for the variable bedrooms.

Answer to Problem 51DA

The bar chart for the number of bedrooms for the variable bedrooms is as follows:

Explanation of Solution

For the variable bedrooms, the frequency table is obtained as follows:

Number of bedrooms	Frequency
2	24
3	26
4 or more	55
Total	105

The bar chart for the given data can be drawn using EXCEL.

Step-by-step procedure to obtain the bar chart using EXCEL is as follows:

• Enter the column of bedrooms along with the frequency column.
• Select the total data range with labels.
• Go to Insert > Charts > bar chart.
• Select the appropriate bar chart.
• Click OK.

From the above bar chart, the highest frequency occurred in the last category, the shape of the distribution is negatively skewed.