91Ó°ÊÓ

Car rental rates per day for a sample of seven Eastern U.S. cities are as follows (The Wall Street Journal, January 16,2004. $$\begin{array}{lc} \text { City } & \text { Daily Rate } \\ \text { Boston } & \$ 43 \\ \text { Atlanta } & 35 \\ \text { Miami } & 34 \\ \text { New York } & 58 \\ \text { Orlando } & 30 \\ \text { Pittsburgh } & 30 \\ \text { Washington, D.C. } & 36 \end{array}$$ a. Compute the mean, variance, and standard deviation for the car rental rates. b. A similar sample of seven Western U.S. cities showed a sample mean car rental rate of \(\$ 38\) per day. The variance and standard deviation were 12.3 and \(3.5,\) respectively. Discuss any difference between the car rental rates in Eastern and Western U.S. cities.

Step by step solution

01

Calculate the Mean for Eastern U.S. Cities

The mean is calculated by summing all the rental rates and then dividing by the number of cities. The rates are as follows: \(43, 35, 34, 58, 30, 30, 36\). Add these rates together: \(43 + 35 + 34 + 58 + 30 + 30 + 36 = 266\). Divide by the number of cities, \(7\): \[\text{Mean} = \frac{266}{7} \approx 38\\]

02

Find the Variance for Eastern U.S. Cities

Variance is calculated by finding the average of the squared differences from the Mean. Each difference from the mean \((38)\) is squared: \[(43 - 38)^2 = 25, \ (35 - 38)^2 = 9, \ (34 - 38)^2 = 16, \ (58 - 38)^2 = 400, \ (30 - 38)^2 = 64, \ (30 - 38)^2 = 64, \ (36 - 38)^2 = 4 \]Add these squared differences: \(25 + 9 + 16 + 400 + 64 + 64 + 4 = 182\). Divide by the number of data points, \(7\):\[\text{Variance} = \frac{182}{7} \approx 26\\]

03

Calculate the Standard Deviation for Eastern U.S. Cities

Standard deviation is the square root of the variance. From Step 2, the variance is \(26\). Therefore, the standard deviation is:\[\text{Standard Deviation} = \sqrt{26} \approx 5.1\]

04

Compare Eastern and Western U.S. Car Rental Rates

The mean car rental rate in the Eastern U.S. cities is \(38\) with a variance of \(26\) and a standard deviation of approximately \(5.1\). For Western U.S. cities, the mean is \(38\), but the variance \(12.3\) and standard deviation \(3.5\) indicate lesser variability among the rates, suggesting more consistency compared to the Eastern U.S. cities.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Mean Calculation

To find the mean of a set of numbers, add all the numbers together and then divide by the count of the numbers.
The mean gives a central value that represents the data set. In this exercise, the mean of car rental rates for Eastern U.S. cities is calculated by adding all seven city rates and dividing by seven.
This calculation gives us an idea of the average daily rate across these cities.

• The formula for Mean is: \( ext{Mean} = \frac{\text{Sum of all rates}}{\text{Number of rates}} \)
• This helps to understand what a typical rental rate looks like across the sampled cities. For this exercise, the mean rate is approximately \(38\).

The mean is important for summarizing data with a single number that reflects the overall rate tendency.

Variance Calculation

Variance measures the spread of numbers in a data set and is calculated by taking the average of the squared differences from the mean.
This tells us how much the rental rates differ from the average rate. A larger variance means more spread out rates.
Here's how it works:

• Subtract the mean from each rate to find the deviation from the mean.
• Square each deviation to eliminate negative values.
• Add up all the squared deviations.
• Divide this total by the number of rates to get the average squared deviation, known as variance.
  The formula is: \( ext{Variance} = \frac{\text{Sum of squared deviations}}{\text{Number of items}} \)

In the exercise, the variance for Eastern U.S. cities came out to be \(26\). This number shows how much individuals rates differ from the average (mean) rate.

Standard Deviation Calculation

Standard deviation is the square root of the variance and provides a measure of the average distance from the mean in the same unit as the original data.
It gives a clearer sense of spread as compared to variance, which is in squared units. The steps to calculate it are simple:

• Start with the variance, which we calculated as the average of squared differences from the mean.
• Take the square root of the variance.
• This gives you the standard deviation: \( ext{Standard Deviation} = \sqrt{\text{Variance}}\)

For the Eastern U.S. cities, the standard deviation is approximately \(5.1\).
This means, on average, each city's rental rate is \(5.1\) away from the mean of \(38\). It's a useful number for understanding the extent of variability.

Comparative Analysis

Comparative analysis involves comparing different datasets to understand differences or similarities.
In this exercise, we compare car rental data from Eastern and Western U.S. cities.
Eastern cities show a mean rental rate similar to Western at \(38\), but their variance and standard deviation are higher.

• Eastern Cities: Variance is \(26\), indicating more variability in rates.
• Western Cities: Variance is \(12.3\), suggesting more consistency.
• Standard deviation for Eastern is \(5.1\) versus \(3.5\) for Western, showing Eastern cities have more variability.

This analysis tells us that while average rates are the same, Eastern cities experience more fluctuation in prices.
Western U.S. cities' rates are more predictable and stable, impacting budgeting and economic planning.