91Ó°ÊÓ

The National Association of Realtors reported the median home price in the United States and the increase in median home price over a five-year period (The Wall Street Journal, January 16,2006 ). Use the sample home prices shown here to answer the following questions. \\[ \begin{array}{llllll} 995.9 & 48.8 & 175.0 & 263.5 & 298.0 & 218.9 & 209.0 \\ 628.3 & 111.0 & 212.9 & 92.6 & 2325.0 & 958.0 & 212.5 \end{array} \\] a. What is the sample median home price? b. In January 2001 , the National Association of Realtors reported a median home price of \(\$ 139,300\) in the United States. What was the percentage increase in the median home price over the five-year period? c. What are the first quartile and the third quartile for the sample data? d. Provide a five-number summary for the home prices. e. Do the data contain any outliers? f. What is the mean home price for the sample? Why does the National Association of Realtors prefer to use the median home price in its reports?

Step by step solution

01

Organize and Sort the Sample Data

First, let's organize and sort the home prices in ascending order: \[\begin{array}{cccccccc} 48.8 & 92.6 & 111.0 & 175.0 & 209.0 & 212.5 & 212.9 & 218.9 & 263.5 & 298.0 & 628.3 & 958.0 & 995.9 & 2325.0 \end{array}\]

02

Calculate the Median Home Price

The median is the middle value of a sorted list. With 14 data points, the median is the average of the 7th and 8th values: \( \text{Median} = \frac{212.9 + 218.9}{2} = 215.9 \).

03

Compute the Percentage Increase

To find the percentage increase from 2001 to 2006: \[ \text{Percentage Increase} = \frac{215.9 - 139.3}{139.3} \times 100 \approx 55.0\% \].

04

Determine the First and Third Quartiles

For 14 data points, the first quartile (Q1) is the 3.75th value, approximated by the 4th value: 175.0. The third quartile (Q3) is the 10.25th value, approximated by the 11th value: 628.3. Hence, \( Q1 = 175.0 \), \( Q3 = 628.3 \).

05

Construct a Five-Number Summary

A five-number summary consists of the minimum, Q1, median, Q3, and maximum: \(48.8, 175.0, 215.9, 628.3, 2325.0\).

06

Identify Outliers using IQR

Calculate the interquartile range (IQR): \( \text{IQR} = 628.3 - 175.0 = 453.3 \). Any data point below \(Q1 - 1.5 \times \text{IQR} = 175.0 - 679.95 = -504.95\) or above \(Q3 + 1.5 \times \text{IQR} = 628.3 + 679.95 = 1308.25\) is an outlier. So, 2325.0 is an outlier.

07

Calculate the Mean Home Price

Add all the home prices and divide by 14: \( \frac{995.9 + 48.8 + 175.0 + 263.5 + 298.0 + 218.9 + 209.0 + 628.3 + 111.0 + 212.9 + 92.6 + 2325.0 + 958.0 + 212.5}{14} = 492.2 \).

08

Discuss Preference of Median over Mean

The National Association of Realtors prefers to use the median because it is not influenced by extremely high or low values (outliers), providing a better central tendency measure. The mean is skewed by outliers like 2325.0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Median

In descriptive statistics, the median is an essential measure of central tendency. Unlike the mean, which sums all values and divides by the number of observations, the median represents the middle point in a dataset. To find the median, you need to arrange the data in ascending order. Once sorted, the middle value is the median. If the dataset has an even number of observations, as in the exercise above with 14 data points, the median is the average of the middle two numbers.
The calculation, in this case, involves the values 212.9 and 218.9. Therefore, the median home price is given by \[ \text{Median} = \frac{212.9 + 218.9}{2} = 215.9 \]
Using the median rather than the mean is often preferred by organizations like the National Association of Realtors because it provides a more accurate representation of the data's center, especially in the presence of outliers.

Outliers

Outliers are data points that differ significantly from other observations. They can distort statistical computations like the mean, hence why understanding and identifying them is crucial. In this exercise, we identify outliers by using the Interquartile Range (IQR) method.
First, calculate IQR by subtracting the first quartile (Q1) from the third quartile (Q3). For our dataset: \[ \text{IQR} = Q3 - Q1 = 628.3 - 175.0 = 453.3\]
Any data point less than \(Q1 - 1.5 \times \text{IQR}\) or more than \(Q3 + 1.5 \times \text{IQR}\) is an outlier. In the example, the calculation shows that a home price of 2325 is an outlier because it is above 1308.25.
Detecting outliers allows analysts to decide whether these points are errors or unusual features of the data that might require special consideration or exclusion from certain analyses.

Five-number summary

A five-number summary provides a quick snapshot of a dataset's distribution. It summarizes data with five key statistics: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
For the home prices in the exercise, the five-number summary is as follows:

• Minimum: 48.8
• First Quartile (Q1): 175.0
• Median: 215.9
• Third Quartile (Q3): 628.3
• Maximum: 2325.0

This summary highlights the spread and central tendency in data, useful for understanding overall trends and identifying outliers. Additionally, it is a standard foundation for creating box plots, providing a visual representation of the dataset's variability.