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Chapter 3: Problem 79

In its recent Fuel Economy Guide, the Environmental Protection Agency gives data on 1152 vehicles. There are a number of outliers, mainly vehicles with very poor gas mileage. If we ignore the outliers, however, the combined city and highway gas mileage of the other 1120 or so vehicles is approximately Normal with mean 18.7 miles per gallon (mpg) and standard deviation 4.3 mpg. The Chevrolet Malibu with a four-cylinder engine has a combined gas mileage of 25 mpg. What percent of all vehicles have worse gas mileage than the Malibu?

Short Answer

Expert verified
92.92% of vehicles have worse gas mileage than the Malibu.

Step by step solution

01

Understand the Normal Distribution

The problem states that the gas mileage of vehicles follows a normal distribution with a mean of 18.7 mpg and a standard deviation of 4.3 mpg. We need to find the percentile rank of the Chevrolet Malibu's mileage of 25 mpg within this distribution.
02

Calculate the Z-score

We use the formula for the Z-score: \[ Z = \frac{X - \mu}{\sigma} \]where \( X = 25 \) mpg is the value for the Malibu, \( \mu = 18.7 \) is the mean, and \( \sigma = 4.3 \) is the standard deviation. This gives us:\[ Z = \frac{25 - 18.7}{4.3} \approx 1.47 \]
03

Use the Z-score to Find the Percentile

Consult a standard normal distribution table or use a calculator to find the probability corresponding to a Z-score of 1.47. This gives us the percentile rank of the given gas mileage, which represents the fraction of the distribution falling below this Z-score.
04

Interpret the Percentile

A Z-score of 1.47 corresponds to approximately 0.9292 or 92.92% in a standard normal distribution. This means about 92.92% of the vehicles have worse gas mileage than the Chevrolet Malibu.

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Key Concepts

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

Z-score calculation
To understand where a specific value stands within a normal distribution, we calculate its Z-score. The Z-score tells us how many standard deviations a particular value is away from the mean. It's a useful statistic because normal distributions have consistent properties regarding the spread of data.
The formula to calculate the Z-score is: \[ Z = \frac{X - \mu}{\sigma} \] Here is what each symbol represents:
  • \( X \) is the value we're interested in (25 mpg for the Chevrolet Malibu in our case).
  • \( \mu \) (mu) is the mean of the distribution, which is 18.7 mpg.
  • \( \sigma \) (sigma) is the standard deviation, which is 4.3 mpg.
Plugging the numbers in, we get a Z-score of approximately 1.47. This score implies that the Malibu's gas mileage is 1.47 standard deviations better than the average car.
Percentile rank
The percentile rank of a value shows the percentage of data points that fall below it in a distribution. It's a way of easily seeing how a specific result compares to the whole set of data.
Once we have the Z-score, finding the percentile rank requires either looking at standard normal distribution tables or using statistical software to determine the proportion of data below our Z-score.
For the Z-score of 1.47, standard tables or calculators tell us this corresponds to approximately 92.92%.
This percentile rank indicates that the Chevrolet Malibu's mileage is better than roughly 92.92% of other vehicles in the study.
Standard deviation
Standard deviation is a measure of the amount of variation or dispersion of a set of values. In a normal distribution, it tells us how tightly or widely the data is spread around the mean.
For this exercise, the standard deviation is 4.3 mpg. This figure provides context for the Z-score and helps us understand how typical or atypical a particular mileage is.
When the standard deviation is small, most data points cluster around the mean. If it's large, the data points are more spread out. The value of 4.3 mpg, therefore, gives us an idea of how much individual vehicle mileages tend to differ from the average mileage of 18.7 mpg.
Mean
The mean is an average that represents the central point or typical value of a dataset. This measure is crucial because it serves as the baseline around which the standard deviation and Z-score are calculated.
In our scenario, the mean gas mileage is 18.7 mpg, which acts as the benchmark for assessing how the Chevrolet Malibu's performance stands relative to other vehicles.
A straightforward calculation of the mean involves summing all values in a dataset and then dividing by the number of values. It helps provide a simple summary and comparison point within the data, giving us insights into whether performance is typical, above, or below average.

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