91Ó°ÊÓ

The Dow Jones Travel Index reported what business travelers pay for hotel rooms per night in major U.S. cities (The Wall Street Joumal, January 16,2004 ). The average hotel room rates for 20 cities are as follows: $$\begin{array}{lclc} \text { Atlanta } & \$ 163 & \text { Minneapolis } & \$ 125 \\ \text { Boston } & 177 & \text { New Orleans } & 167 \\ \text { Chicago } & 166 & \text { New York } & 245 \\ \text { Cleveland } & 126 & \text { Orlando } & 146 \\ \text { Dallas } & 123 & \text { Phoenix } & 139 \\ \text { Denver } & 120 & \text { Pittsburgh } & 134 \\ \text { Detroit } & 144 & \text { San Francisco } & 167 \\ \text { Houston } & 173 & \text { Seattle } & 162 \\ \text { Los Angeles } & 160 & \text { St. Louis } & 145 \\ \text { Miami } & 192 & \text { Washington, D.C. } & 2077\end{array}$$ a. What is the mean hotel room rate? b. What is the median hotel room rate? c. What is the mode? d. What is the first quartile? c. What is the third quartile?

Step by step solution

01

Organize Data

List the hotel rates in ascending order: 120, 123, 125, 126, 134, 139, 144, 145, 146, 160, 162, 163, 166, 167, 167, 173, 177, 192, 207, 245.

02

Calculate the Mean

Add all hotel rates: \[120 + 123 + 125 + 126 + 134 + 139 + 144 + 145 + 146 + 160 + 162 + 163 + 166 + 167 + 167 + 173 + 177 + 192 + 207 + 245 = 3157\]. Divide the total by 20 (the number of cities): \[\frac{3157}{20} = 157.85\].

03

Determine the Median

Since there are 20 data points, the median is the average of the 10th and 11th values. The 10th and 11th values in the ordered list are 160 and 162. Calculate the median: \[\frac{160 + 162}{2} = 161\].

04

Identify the Mode

The mode is the number that appears most frequently. In the list, 167 appears twice. Therefore, 167 is the mode.

05

Calculate the First Quartile

The first quartile (Q1) is the median of the first half of the data. The first 10 data points are: 120, 123, 125, 126, 134, 139, 144, 145, 146, 160. The median of these numbers is \[\frac{134 + 139}{2} = 136.5\].

06

Calculate the Third Quartile

The third quartile (Q3) is the median of the second half of the data. The last 10 data points are: 162, 163, 166, 167, 167, 173, 177, 192, 207, 245. The median of these numbers is \[\frac{167 + 173}{2} = 170\].

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Key Concepts

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

Mean Calculation

The mean, commonly known as the average, is a fundamental concept in descriptive statistics. It provides a central value for a set of numbers. Calculating the mean involves summing all the data points and then dividing by the count of the data points.

In the context of our hotel rate problem, we first list all nightly rates and sum them up: 120, 123, 125, 126, 134, 139, 144, 145, 146, 160, 162, 163, 166, 167, 167, 173, 177, 192, 207, 245, which totals to 3157. Next, we divide by the number of cities, which is 20. So, the mean is calculated as:

• Add all the rates: \( 3157 \)
• Divide by the number of data points: \( \frac{3157}{20} = 157.85 \)

Therefore, the mean hotel room rate is approximately $157.85.

Median Calculation

The median serves as the middle value in a data set and is particularly useful when dealing with skewed data.

To find the median, we first need to arrange our data points in ascending order. Our list is already ordered: 120, 123, 125, 126, 134, 139, 144, 145, 146, 160, 162, 163, 166, 167, 167, 173, 177, 192, 207, 245. With 20 numbers in total, the median will be the average of the 10th and 11th numbers. These numbers are 160 and 162:

• Add the 10th and 11th values: \( 160 + 162 = 322 \)
• Divide by 2: \( \frac{322}{2} = 161 \)

Thus, the median hotel room rate is $161.

Quartiles

Quartiles divide the data into four equal parts, giving insights on the spread and distribution.

**First Quartile (Q1)**:
To find the first quartile, we need the median of the first half of the ordered data. The first 10 points are 120, 123, 125, 126, 134, 139, 144, 145, 146, 160. The median is between 134 and 139:

• Add 134 and 139: \( 134 + 139 = 273 \)
• Divide by 2: \( \frac{273}{2} = 136.5 \)

**Third Quartile (Q3)**:
This is the median of the second half of the data, which includes: 162, 163, 166, 167, 167, 173, 177, 192, 207, 245. The median of these is between 167 and 173:

• Add 167 and 173: \( 167 + 173 = 340 \)
• Divide by 2: \( \frac{340}{2} = 170 \)

The first quartile is approximately 136.5 and the third quartile is 170.

Data Distribution

Understanding how data is distributed is essential for identifying patterns and making inferences.

When we organize the data, it becomes easier to spot how values cluster or spread. This dataset showcases the range of hotel rates from $120 to $245. By examining the data set:

• Determine the spread by subtracting the smallest from the largest value: 245 - 120 = 125
• Cluster near the median, with both halves of the data having similar spreads
• Outliers, like the $245 rate, might affect the mean more than the median

The organized list and measures like the mean, median, and quartiles help paint a clear picture of the rate distribution across cities.

Mode Determination

The mode is the most frequently occurring value in a data set, providing insight into the most common data point.

For our hotel room rate data, we observe each rate to find any repeats. The list shows that 167 occurs twice, which is more frequent than any other rate. Therefore, the mode of this data set is 167.

Understanding the mode helps identify the typical rate that appears most often, which can be especially useful in settings where you want to find the most common occurrence rather than just the average or middle.