Problem 28 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

TRAFFIC MANAGEMENT Suppose the random variable \(X\) in Exercise 27 is normally distributed with mean \(\mu=12\) feet and standard deviation \(\sigma=4\) feet. Now what is the probability that a randomly selected pair of cars will be less than 10 feet apart?

Step-by-Step Solution

Answer

The probability is 30.85%.

Step 1: Define the problem

We need to find the probability that a randomly selected pair of cars will be less than 10 feet apart, given that the distance between cars (\(X\)) is normally distributed with \(\mu=12\) feet and 🌸\(\sigma=4\) feet.

Step 2: State the known values

The mean (\(\mu\)) is 12 feet and the standard deviation (\(\sigma\)) is 4 feet. We need to find \(P(X < 10)\).

Step 3: Calculate the Z-score

The Z-score formula is \(Z = \frac{X - \mu}{\sigma}\). Here, \(X = 10\), so the Z-score for 10 feet is: \[Z = \frac{10 - 12}{4} = \frac{-2}{4} = -0.5\]

Step 4: Find the cumulative probability

Using the Z-score of -0.5, look up the corresponding cumulative probability in the standard normal distribution table. The value for \(Z = -0.5\) is approximately 0.3085.

Step 5: Interpret the result

The probability that a randomly selected pair of cars will be less than 10 feet apart is 0.3085, or 30.85%.

Key Concepts

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by its bell-shaped curve.
This curve is symmetric around the mean, which is the highest point on the graph.
The total area under the curve is 1, representing the total probability of all outcomes.
The equation for the normal distribution is given by:

Z-score

The Z-score, also known as the standard score, is a measure of how many standard deviations an element is from the mean.
It is calculated using the formula:

Cumulative Probability

Cumulative probability is the probability that a random variable will take a value less than or equal to a given value.
It is found using the cumulative distribution function (CDF) of the normal distribution.
The CDF gives the area under the curve to the left of a given value.
This area represents the cumulative probability.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values.
It is denoted by the symbol \( \sigma \). In a normal distribution, the standard deviation determines the width of the bell curve.

Mean

The mean, also known as the average, is the sum of all the values in a set divided by the number of values.
It is denoted by the symbol \( \mu \). In a normal distribution, the mean is the center of the symmetry of the bell curve.
Therefore, the mean is one of the key parameters that define the normal distribution.