Problem 35 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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 \(50+c .\)

Step-by-Step Solution

Answer

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 1: 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: \[0.96 \leq 1 - \frac{1}{k^2}.\]

Step 2: 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.

Step 3: 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 \(7\).

Key Concepts

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.