Problem 214 - Applied Statistics and Probability for Engineers

Question

The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours. (a) What is the probability that a laser fails before 5800 hours? (b) What is the life in hours that \(90 \%\) of the lasers exceed? (c) What should the mean life equal for \(99 \%\) of the lasers to exceed 10,000 hours before failure? (d) A product contains three lasers, and the product fails if any of the lasers fails. Assume that the lasers fail independently. What should the mean life equal for \(99 \%\) of the products to exceed 10,000 hours before failure?

Step-by-Step Solution

Answer

(a) 2.28%, (b) 6232 hours, (c) 11400 hours, (d) 8350 hours.

Step 1: Set Up for Part (a)

Part (a) requires finding the probability that a laser fails before 5800 hours. We need the Z-score that corresponds to 5800 hours using the formula:\[ Z = \frac{X - \mu}{\sigma} \]where \( X \) is 5800, \( \mu \) is the mean (7000 hours), and \( \sigma \) is the standard deviation (600 hours).

Step 2: Calculate Z-Score for Part (a)

Calculate the Z-score:\[ Z = \frac{5800 - 7000}{600} = \frac{-1200}{600} = -2.0 \]

Step 3: Probability for Part (a)

Using a standard normal distribution table or calculator, find the probability corresponding to \( Z = -2.0 \). This probability is approximately 0.0228. Hence, the probability that a laser fails before 5800 hours is 2.28%.

Step 4: Set Up for Part (b)

Part (b) requires finding the life in hours that 90% of the lasers exceed. This means we need to find the 10th percentile of the normal distribution.

Step 5: Use Z-Score for Part (b)

Consult a standard normal distribution table to find the Z-score that results in 0.10 in the left tail. The Z-score is approximately -1.28.

Step 6: Calculate Life in Hours for Part (b)

Using the Z-score, solve the following equation for \( X \):\[ -1.28 = \frac{X - 7000}{600} \]Solving gives:\[ X = 7000 + (-1.28 \times 600) = 6232 \]Thus, 90% of the lasers exceed 6232 hours.

Step 7: Set Up for Part (c)

Part (c) requires finding the mean life for 99% of lasers to exceed 10,000 hours. We first find the Z-score for 1% left tail with a value of approximately -2.33.

Step 8: Calculate Mean for Part (c)

Using the given formula and solving for the mean:\[ -2.33 = \frac{10000 - \mu}{600} \]Solving gives:\[ \mu = 10000 + (2.33 \times 600) = 11400 \]

Step 9: Set Up for Part (d)

Part (d) requires all three lasers in the product to surpass 10,000 hours for it to not fail. We want 99% products to exceed this, so the probability of each laser not failing is the cube root of 0.99.

Step 10: Calculate Mean for Part (d)

Let \( p \) be the probability that each laser doesn't fail. Solve \( p^3 = 0.99 \), so \( p = 0.9967 \). Using the Z-score for 0.9967 which is approximately 2.75:\[ 2.75 = \frac{10000 - \mu}{600} \]Solving gives:\[ \mu = 10000 - (2.75 \times 600) = 8350 \]

Key Concepts

Normal Distribution

In probability theory, the normal distribution is one of the most fundamental and commonly used continuous probability distributions. It is sometimes known as the Gaussian distribution. The distinctive bell-shaped curve is symmetrical around the mean, with its shape defined by the mean (\(\mu\)) and standard deviation (\(\sigma\)).This distribution is essential because it describes how data values tend to pile up around a central value due to natural phenomena. Many natural and social phenomena tend to display this type of distribution, such as heights, grades, intelligence scores, and, in our exercise context, the life span of semiconductor lasers. The central role of the normal distribution is supported by the Central Limit Theorem, which states that sums of random samples tend to follow a normal distribution, regardless of the underlying distribution.

• Symmetrical curve: The graph is a perfect bell shape, having maximum frequency at the center.
• Defined by mean and standard deviation: The mean dictates the center of the graph, while the standard deviation controls its width.
• Area under the curve equals 1: As the total probability is 1, the area under the curve equals 1 as well.

Understanding this distribution allows us to assess probabilities and outcomes related to the exercise, such as calculating the likelihood of a laser failing before a given number of hours.

Z-score

The Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. When data follows a normal distribution, the Z-score helps determine the position of a particular value relative to the mean. It's calculated using \[ Z = \frac{X - \mu}{\sigma} \] where\( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.The Z-score represents how many standard deviations a particular value is from the mean:

• A Z-score of 0 indicates the value is exactly at the mean.
• Positive Z-scores indicate values above the mean.
• Negative Z-scores indicate values below the mean.

Understanding Z-scores is vital because they allow us to work with various values on a standardized scale, facilitating comparison and probability calculations. For example, as seen in the exercise, determining the Z-score of a laser's life span below a certain number of hours helps calculate the probability of such an event happening.

Percentiles

Percentiles indicate the relative standing of an observation in a data set, showing the percentage of data that lies below a particular value. In other words, the nth percentile is the value below which n percent of observations fall. In the context of a normal distribution, percentiles help us interpret the distribution of data more intuitively. For example:

• The 50th percentile (median) divides the data into two equal halves.
• The 10th percentile indicates that 10% of data is below it and 90% is above it.
• The 90th percentile shows the value below which 90% of the data falls.

Percentiles are critical in finding outliers or understanding the spread and saturation of a dataset. In our exercise, determining the life in hours that 90% of lasers exceed requires calculating the 10th percentile of the normal distribution. It involves using a Z-score to determine the cut-off point at which only 10% of the life spans lie below a certain number of hours.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena and quantifying the likelihood of occurrences. It provides the framework that allows us to model uncertainties and make predictions about various outcomes. Some fundamental concepts of probability theory include:

• Events: Outcomes from a random phenomenon, where probability is assigned.
• Random Variables: Variables that take on different values depending on outcomes of a random event.
• Probability Distributions: Mathematical functions that provide probabilities of occurrence of different possible outcomes.

In the exercise, probability theory helps us assess how likely it is for a semiconductor laser to fail within a specified time. By applying normal distribution concepts and calculating Z-scores, we can find probabilities tied to time-life thresholds. The calculations allow us to measure risk and performance and help in planning mean life adjustments for lasers to ensure desired operational levels are met.