A car battery has an average lifetime of 48 months with a...

A car battery has an average lifetime of 48 months with a standard deviation of 6 months. The battery lives are normally distributed. The probability that a given battery will last between 48 months and 60 months is $0.0665 \int_{48}^{60} e^{-0.0139(t-48)^{2}} d t .$ Use the integration capabilities of a graphing utility to approximate the integral. Interpret the resulting probability.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by two parameters: the mean, which indicates the location of the center of the distribution, and the standard deviation, which measures the spread of the data. In many real-world scenarios, such as the lifetime of a product, the normal distribution provides a useful model for natural variation.

Probability Density Function (PDF)

The probability density function (PDF) for a continuous random variable describes the relative likelihood of the variable taking on a specific value. In the case of the normal distribution, the PDF determines the probability of the variable falling within a certain interval, with the area under the curve between two points representing that probability.

Numerical Integration

Numerical integration is a method used to approximate the value of an integral when an analytical solution is difficult or impossible to obtain. Graphing utilities and computational tools can perform numerical integration, providing approximate values for areas under curves, which is particularly useful in probability calculations involving continuous distributions.

Interpretation of Probability

In the context of probability, the integral of the PDF over an interval represents the probability that the random variable will take a value within that range. For instance, when calculating the integral from 48 to 60, the resulting value is the probability that a battery will last between these two time points, offering a concrete measure of how likely that event is within the given distribution.

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