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

List the Car Rental Rates

First, extract the daily rental rates from the provided table for each city: \( [43, 35, 34, 58, 30, 30, 36] \).

02

Calculate the Mean

The mean is found by dividing the sum of all rates by the number of rates. Sum the rates: \( 43 + 35 + 34 + 58 + 30 + 30 + 36 = 266 \). Divide by 7: \( \frac{266}{7} = 38 \). So, the mean is \( 38 \).

03

Calculate Variance

Variance is calculated by averaging the squared differences from the mean. First, subtract the mean from each rate and square the result: \([ (43-38)^2, (35-38)^2, (34-38)^2, (58-38)^2, (30-38)^2, (30-38)^2, (36-38)^2 ] = [25, 9, 16, 400, 64, 64, 4] \). Sum these: \( 25 + 9 + 16 + 400 + 64 + 64 + 4 = 582 \). Divide by 6 (since it's a sample and not the whole population): \( \frac{582}{6} \approx 97 \). The variance is \( 97 \).

04

Calculate Standard Deviation

The standard deviation is the square root of the variance. So, the standard deviation is \( \sqrt{97} \approx 9.85 \).

05

Compare Eastern and Western Rates

For the Western cities, the mean was already given as 38, which is the same as the Eastern cities. The variance in Eastern cities (97) is higher than the Western (12.3), suggesting greater variability in Eastern rates. The standard deviation of the Eastern cities (≈9.85) is also higher than the Western (3.5), reinforcing that Eastern rates vary more from the average.

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Key Concepts

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

Mean Calculation

The mean, often referred to as the average, is a measure that represents the central point of a dataset. If you want to know the typical car rental rate for a group of cities, calculating the mean gives you a good idea.
To find the mean for the car rental rates in the provided list of cities, you would follow these simple steps:

• Add all the daily rental rates together.
• For our example: Boston ( $43 ), Atlanta ( $35 ), Miami ( $34 ), New York ( $58 ), Orlando ( $30 ), Pittsburgh ( $30 ), Washington D.C. ( $36 ).
• The total is $266 .
• Divide this total by the number of cities, which is 7.

This gives you a mean car rental rate of $38 per day. The mean value tells you where most car rental rates center around, successfully summarizing them with a single number which is easy to understand.

Variance

Variance represents how much the values in a dataset differ from the mean. It gives you an idea of the spread or dispersion of the data.
In our example, we find that each rate differs from the mean $38 , so we look at these differences:

• First, find the difference between each city's rate and the mean, then square those differences.
• For example: for Boston, (43 - 38)^2 = 25 ; for Atlanta, (35 - 38)^2 = 9 ; and so on till Washington, D.C.
• The squared differences are [25, 9, 16, 400, 64, 64, 4] .
• Then, sum these squared differences to get $582 .
• Finally, divide by the number of rates minus one ( 6 ) to get the variance ( 97 ).

The variance of $97 indicates how spread out the car rental rates are from the mean. A higher variance means more spread, whereas a smaller variance indicates that the rates are closer to the mean.

Standard Deviation

Standard deviation is another measure of how spread out the numbers in your set of data are; it is closely related to variance. While variance is in squared units, the standard deviation brings it back to the original units by taking the square root. This makes it easier to understand and compare.

• For the car rentals, after calculating the variance as \(97, take the square root.
• Using a calculator or some quick maths: \( \sqrt{97} \approx 9.85 \).
• This tells us that the typical variability or spread from the mean (\)38 per day) is about $9.85.

In practice, the standard deviation allows us to assess whether most of the rates are clustered near the mean or if there are big differences in the rates. A smaller standard deviation means prices are more consistent, while a larger one suggests more variation among the rates.