Problem 22 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

PRODUCT RELIABILITY Electronic components made by a certain process have a time to failure that is measured by a normal random variable. If the mean time to failure is 15,000 hours with a standard deviation of 800 hours, what is the probability that a randomly selected component will last no more than 10,000 hours?

Step-by-Step Solution

Answer

The probability that a component lasts no more than 10,000 hours is approximately 0.

Step 1: Understand the Problem

The exercise asks for the probability that a component lasts no more than 10,000 hours given a normal distribution with a mean of 15,000 hours and a standard deviation of 800 hours.

Step 2: Identify the Variables

The mean \(\text{μ}\) is 15,000 hours and the standard deviation \(\text{σ}\) is 800 hours. We need to find the probability that \(X \leq 10,000\) hours.

Step 3: Convert to Standard Normal Distribution

Use the z-score formula to convert 10,000 hours to a standard normal variable: \[ z = \frac{X - \text{μ}}{\text{σ}} = \frac{10,000 - 15,000}{800} = \frac{-5,000}{800} = -6.25 \]

Step 4: Find the Probability

Use the z-table to find the probability corresponding to the z-score \(-6.25\). Since \(-6.25\) is a very extreme value, the probability is practically 0.

Key Concepts

normal distribution

The normal distribution is a common probability distribution in statistics. It's often called the bell curve because of its shape. This distribution is symmetric around the mean, meaning it looks the same to the left and right of the center point.

• Characteristics: The normal distribution is described by its mean (average) and standard deviation (spread or width).
• Mean (μ): This is the peak's center.

Most data falls within three standard deviations from the mean.
The total area under the curve is 1, representing the total probability.
In this exercise, the time to failure of electronic components is normally distributed with a mean (μ) of 15,000 hours and a standard deviation (σ) of 800 hours.

mean and standard deviation

To understand how long components last before failure, it's crucial to know the mean and standard deviation.

• Mean (μ): This is the average time until failure, 15,000 hours. It indicates the center of the data.
• Standard Deviation (σ): This measures how spread out the data is. Here, it's 800 hours. A small standard deviation means the times are close to the mean, while a large one indicates a wide range of times.

These two values help us understand the general performance and reliability of the components. In this context, they tell us how long components can be expected to last and how much variation there is in those times.

z-score

The z-score is a way to translate a value from a normal distribution to a standard normal distribution, which has a mean of 0 and a standard deviation of 1. It's a measure of how many standard deviations an element is from the mean.
In the exercise:

• Calculation: The z-score is found with the formula: \[ z = \frac{X - \text{μ}}{\text{σ}} \]
• Example: For a component with a failure time of 10,000 hours: \[ z = \frac{10,000 - 15,000}{800} = -6.25 \]
  This score tells us how far 10,000 hours is from the mean in terms of standard deviations.
• Interpretation: We use this z-score to find the probability of a component failing by 10,000 hours. Z-scores also help us understand the rarity of an event in the context of a normal distribution.

In this example, the z-score of -6.25 is very extreme, suggesting that it’s very unlikely a component will fail at or before 10,000 hours.