Question

An automobile manufacturer introduces a new model that averages 27 miles per gallon in the city. A person who plans to purchase one of these new cars wrote the manufacturer for the details of the tests, and found out that the standard deviation is 3 miles per gallon. Assume that in-city mileage is approximately normally distributed. a. What is the probability that the person will purchase a car that averages less than 20 miles per gallon for in-city driving? b. What is the probability that the person will purchase a car that averages between 25 and 29 miles per gallon for in-city driving?

Short answer

a) 0.99% probability for less than 20 mpg. b) 49.72% probability for between 25 and 29 mpg.

Step-by-step solution

Understanding the Problem

We are given a normally distributed random variable representing miles per gallon (mpg) for in-city driving with a mean \( \mu = 27 \) mpg and a standard deviation \( \sigma = 3 \) mpg. We need to find probabilities for certain mileage intervals.

Calculate the Z-score for Part (a)

For part (a), we determine the Z-score for \( x = 20 \) mpg using the formula: \[ Z = \frac{x - \mu}{\sigma} = \frac{20 - 27}{3} = \frac{-7}{3} = -2.33 \]

Compute the Probability for Part (a)

Using standard normal distribution tables or a calculator, find the probability that Z is less than -2.33. This gives us the probability that a car averages less than 20 mpg: \[ P(X < 20) = P(Z < -2.33) \approx 0.0099 \] Thus, the probability is approximately 0.99%.

Calculate the Z-scores for Part (b)

For part (b), compute the Z-scores for \( x = 25 \) mpg and \( x = 29 \) mpg: 1. For 25 mpg: \[ Z_{25} = \frac{25 - 27}{3} = \frac{-2}{3} = -0.67 \]2. For 29 mpg:\[ Z_{29} = \frac{29 - 27}{3} = \frac{2}{3} = 0.67 \]

Compute the Probabilities for Part (b)

Using a standard normal distribution table or a calculator: 1. Find \( P(Z < 0.67) \approx 0.7486 \) 2. Find \( P(Z < -0.67) \approx 0.2514 \)3. Compute the probability between 25 and 29 mpg: \[ P(25 < X < 29) = P(Z < 0.67) - P(Z < -0.67) = 0.7486 - 0.2514 = 0.4972 \] Thus, the probability is approximately 49.72%.

Key concepts

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric around its mean, depicting a bell-shaped curve. It is one of the most important probability distributions in statistics, as many real-world phenomena approximate this distribution.

• A normal distribution is defined by its mean (average) and standard deviation. These parameters determine the position and spread of the distribution curve.
• The mean is the peak of the curve and determines the location of the center. As a result, more values are likely to occur closer to the mean.
• The standard deviation affects the width of the distribution. A smaller standard deviation means the values are more tightly clustered around the mean, whereas a larger standard deviation means they are more spread out.

An important property is that around 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. This key characteristic helps in calculating probabilities, as seen in the solution above.

Z-Score

The Z-score is a measure that describes a value's relation to the mean of a group of values. In a normal distribution, it indicates how many standard deviations an element is from the mean.

• The formula for calculating the Z-score is: \[ Z = \frac{x - \mu}{\sigma} \] where \( x \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
• A Z-score of 0 represents a value equal to the mean. A positive Z-score indicates a value above the mean, whereas a negative Z-score indicates a value below the mean.
• Using Z-scores, we can compare different data points across different normal distributions or assess probabilities by referring to standard normal distribution tables.

In the given exercise, Z-scores help us calculate the probability of finding cars with specific mileage, showing how far these values are from the average performance measure of 27 miles per gallon.

Standard Deviation

Standard deviation is a critical statistic that quantifies the amount of variation or dispersion in a set of data values. It helps us understand how spread out the data points are from the mean.

• A small standard deviation means that data points tend to be close to the mean.
• A large standard deviation means that data points are spread out over a wider range.
• It is calculated as the square root of the variance and is expressed in the same units as the data.

For the automobile example, the standard deviation is 3 miles per gallon, indicating how much individual car mileage results vary from the 27 mpg average. By utilizing this measure, you can understand the consistency of the car's mileage performance across different trials.

Mean

The mean, commonly referred to as the average, provides a measure of central tendency, summarizing data into a single value that describes the center.

• It is calculated by summing all the values in a dataset and then dividing by the number of values.
• The mean is sensitive to extreme values (outliers), which can greatly influence its value.
• Because of this sensitivity, while the mean is a useful measure, it may not always accurately reflect the typical value, especially if the data set has significant outliers or skewness.

In the example problem, the mean of 27 miles per gallon represents the expected performance for city driving in the new car model, providing a baseline for comparing actual performance.