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Chapter 8: Problem 35

A probability distribution has a mean of 50 and a standard deviation of \(1.4\). Use Chebychev's inequality to find the value of \(c\) that guarantees the probability is at least \(96 \%\) that an outcome of the experiment lies between \(50-c\) and \(50+c .\)

Short Answer

Expert verified
The value of \(c\) that guarantees the probability is at least \(96 \%\) for an outcome of the experiment to lie between \(50 - c\) and \(50 + c\) is \(7\).

Step by step solution

01

Relate the known probability to Chebyshev's inequality

Given that we want a probability of at least \(96\%\), we can translate that into the inequality: \[ P(50 - c \leq X \leq 50 + c) \geq 0.96 .\] By comparing this to Chebyshev's inequality: \[P(\mu - k\sigma \leq X \leq \mu + k\sigma) \geq 1 - \frac{1}{k^2},\] we can find the relationship between the \(k\) we want and the minimum probability required: \[0.96 \leq 1 - \frac{1}{k^2}.\]
02

Solve for k

To find the value of \(k\), we'll solve the inequality we obtained in step 1: \[0.96 \leq 1 - \frac{1}{k^2}.\] Subtract \(0.96\) from both sides: \[0 \leq 0.04 - \frac{1}{k^2}.\] Add \(\frac{1}{k^2}\) to both sides: \[\frac{1}{k^2} \geq 0.04.\] Now, take the reciprocal of both sides: \[k^2 \leq \frac{1}{0.04}.\] \[k^2 \leq 25.\] Take square root of both sides: \[k \leq 5.\] Since we want the smallest \(k\) that satisfies this inequality, we find that \(k = 5\) by inspection.
03

Calculate the value of c using k value and standard deviation

With \(k = 5\) and a known standard deviation \(\sigma = 1.4\), we can find the value of \(c\) as: \[c = k\cdot\sigma\] \[c = 5 \cdot 1.4\] \[c = 7\] So the value of \(c\) that guarantees the probability is at least \(96 \%\) for an outcome of the experiment to lie between \(50 - c\) and \(50 + c\) is \(7\).

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Key Concepts

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

Probability Distribution
Understanding the concept of a probability distribution is key to grasping many complex ideas in statistics and probability theory. At its simplest, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. For a discrete random variable, this can often be expressed as a list or table, while for a continuous random variable, it takes the form of a curve known as the probability density function.
Standard Deviation
The standard deviation is a measure of how spread out the numbers in a data set are. In probability and statistics, it plays a crucial role as it quantifies the amount of variation or dispersion from the average (mean), and it is defined as the square root of the variance.

The smaller the standard deviation, the closer the data points tend to be to the mean, or expected value. Conversely, a larger standard deviation indicates a wider spread of values. The standard deviation is key in risk assessment and in various fields of science and finance as a means of measuring unpredictability or volatility.
Probability Theory
Probability theory is a branch of mathematics concerned with analyzing random phenomena. The fundamental components of probability theory are experiments, outcomes, and events.

An 'experiment' refers to a procedure that can be infinitely repeated and has a well-defined set of outcomes, 'outcome' is a possible result of an experiment, and an 'event' is a set of outcomes resulting from an experiment. Probability theory provides the framework to predict and understand the likelihood of these events. It is a foundation for statistics and is used in various fields such as finance, science, and engineering.

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Most popular questions from this chapter

Based on past experience, the manager of the VideoRama Store has compiled the following table, which gives the probabilities that a customer who enters the VideoRama Store will buy \(0,1,2,3\), or 4 DVDs. How many DVDs can a customer entering this store be expected to buy? $$\begin{array}{lccccc} \hline \text { DVDs } & 0 & 1 & 2 & 3 & 4 \\\\\hline \text { Probability } & .42 & .36 & .14 & .05 & .03 \\ \hline\end{array}$$

The odds against an event \(E\) occurring are 2 to \(3 .\) What is the probability of \(E\) not occurring?

The management of MultiVision, a cable TV company, intends to submit a bid for the cable television rights in one of two cities, \(A\) or \(B\). If the company obtains the rights to city A, the probability of which is .2, the estimated profit over the next \(10 \mathrm{yr}\) is $$\$ 10$$ million: if the company obtains the rights to city \(\mathrm{B}\), the probability of which is \(.3\), the estimated profit over the next 10 yr is $$\$ 7$$ million. The cost of submitting a bid for rights in city \(\mathrm{A}\) is $$\$ 250,000$$ and that in city B is $$\$ 200,000$$. By comparing the expected profits for each venture, determine whether the company should bid for the rights in city A or city B.

The interest rates paid by 30 financial institutions on a certain day for money market deposit accounts are shown in the accompanying table: $$\begin{array}{lcccc}\hline \text { Rate, \% } & 6 & 6.25 & 6.55 & 6.56 \\ \hline \text { Institutions } & 1 & 7 & 7 & 1 \\ \hline\end{array}$$ $$\begin{array}{lcccc} \hline \text { Rate, } \% & 6.58 & 6.60 & 6.65 & 6.85 \\\\\hline \text { Institutions } & 1 & 8 & 3 & 2 \\ \hline\end{array}$$ Let the random variable \(X\) denote the interest rate paid by a randomly chosen financial institution on its money market deposit accounts and find the probability distribution associated with these data.

A panel of 50 economists was asked to predict the average prime interest rate for the upcoming year. The results of the survey follow: $$\begin{array}{lcccccc}\hline \text { Interest Rate, } \% & 4.9 & 5.0 & 5.1 & 5.2 & 5.3 & 5.4 \\\\\hline \text { Economists } & 3 & 8 & 12 & 14 &8 & 5 \\ \hline\end{array}$$ Based on this survey, what does the panel expect the average prime interest rate to be next year?
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