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Chapter 11: Problem 28

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?

Short Answer

Expert verified
The probability is 30.85%.

Step by step solution

01

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 馃尭\(\sigma=4\) feet.
02

State the known values

The mean (\(\mu\)) is 12 feet and the standard deviation (\(\sigma\)) is 4 feet. We need to find \(P(X < 10)\).
03

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: \[Z = \frac{10 - 12}{4} = \frac{-2}{4} = -0.5\]
04

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.
05

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%.

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Key Concepts

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

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.

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Most popular questions from this chapter

EFFECT OF AN EPIDEMIC A study of the effect of an epidemic of mononucleosis (mono) on the students at a particular small private college determines that during the 30 days of November, there were: 5 days when no students had mono 7 days when exactly one student had mono 4 days when exactly two students had mono 9 days when exactly six students had mono 3 days when exactly seven students had mono 2 days when exactly eight students had mono Let \(X\) be the random variable that measures the number of students with mononucleosis on a randomly selected day in November. a. Find the probability distribution for \(X\). Then construct a histogram for this distribution. b. How many students would you expect to have mononucleosis on a randomly selected day in November?

TIME MANAGEMENT A shuttle tram arrives at a tram stop at a randomly selected time \(X\) within a 1-hour period, and a tourist independently arrives at the same stop also at a randomly selected time \(Y\) within the same hour. The tourist has the patience to wait for the tram for up to 20 minutes before calling a taxi. The joint probability density function for \(X\) and \(Y\) is $$ f(x, y)= \begin{cases}1 & \text { if } 0 \leq x \leq 1,0 \leq y \leq 1 \\ 0 & \text { otherwise }\end{cases} $$ a. What is the probability that the tram takes longer than 20 minutes to arrive? b. What is the probability that the tourist arrives after the tram? c. What is the probability that the tourist connects with the tram? [Hint: The event of this occurring has the form $$ Y+a \leq X \leq Y+b $$ for suitable numbers \(a\) and \(b\).]

MEDICINE Suppose that the number of children who die each year from leukemia follows a Poisson distribution and that on average, \(7.3\) children per 100,000 die from leukemia. For a city with 100,000 children, find the probability of each of the following events: a. Exactly seven children in the city die from leukemia each year. b. Fewer than two children in the city die from leukemia each year. c. More than five children in the city die from leukemia each year.

SPORTS MEDICINE Suppose that the number of injuries a team suffers during a typical football game follows a Poisson distribution with an average of \(2.5\) injuries. a. Find the probability that during a randomly chosen game, the team suffers exactly two injuries. b. Find the probability that during a randomly chosen game, the team suffers no injuries. c. Find the probability that during a randomly chosen game, the team suffers at least one injury.

INSURANCE POLICY An insurance company charges \(\$ 10,000\) for a policy insuring against a certain kind of accident and pays \(\$ 100,000\) if the accident occurs. Suppose it is estimated that the probability of the accident occurring is \(p=0.02\). Let \(X\) be the random variable that measures the insurance company's profit on each policy it sells. a. What is the probability distribution for \(X\) ? b. What is the company's expected profit per policy sold? c. What should the company charge per policy to double its expected profit per policy?
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