Problem 55 - Applied Statistics and Probability for Engineers

Question

In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate. The thickness of these layers is critical if specifications regarding the final color and intensity of light are to be met. Let \(X\) and \(Y\) denote the thickness of two different layers of ink. It is known that \(X\) is normally distributed with a mean of 0.1 millimeter and a standard deviation of 0.00031 millimeter, and \(Y\) is normally distributed with a mean of 0.23 millimeter and a standard deviation of 0.00017 millimeter. The value of \(\rho\) for these variables is equal to \(0 .\) Specifications call for a lamp to have a thickness of the ink corresponding to \(X\) in the range of 0.099535 to 0.100465 millimeter and \(Y\) in the range of 0.22966 to 0.23034 millimeter. What is the probability that a randomly selected lamp will conform to specifications?

Step-by-Step Solution

Answer

The probability is approximately 0.8268, or 82.68%.

Step 1: Calculate Z-scores for X

The thickness variable \(X\) is normally distributed with a mean \(\mu_X = 0.1\) mm and a standard deviation \(\sigma_X = 0.00031\) mm. To find the probability that \(X\) is within the specified range of [0.099535, 0.100465], we first calculate the Z-scores for these limits using the formula:\[ Z = \frac{X - \mu}{\sigma} \]For the lower limit (0.099535 mm):\[ Z_{\text{lower}} = \frac{0.099535 - 0.1}{0.00031} = \frac{-0.000465}{0.00031} \approx -1.5 \]For the upper limit (0.100465 mm):\[ Z_{\text{upper}} = \frac{0.100465 - 0.1}{0.00031} = \frac{0.000465}{0.00031} \approx 1.5 \]

Step 2: Calculate Probability for X

The Z-scores found are \(Z_{\text{lower}} \approx -1.5\) and \(Z_{\text{upper}} \approx 1.5\). Using the standard normal distribution table, we find the probabilities:- \(P(Z \leq -1.5) \approx 0.0668\)- \(P(Z \leq 1.5) \approx 0.9332\)Thus, the probability that \(X\) is within the specified range is:\[ P(-1.5 < Z < 1.5) = P(Z \leq 1.5) - P(Z \leq -1.5) = 0.9332 - 0.0668 = 0.8664 \]

Step 3: Calculate Z-scores for Y

The thickness variable \(Y\) is normally distributed with a mean \(\mu_Y = 0.23\) mm and a standard deviation \(\sigma_Y = 0.00017\) mm. For the specified range of [0.22966, 0.23034], calculate the Z-scores:For the lower limit (0.22966 mm):\[ Z_{\text{lower}} = \frac{0.22966 - 0.23}{0.00017} = \frac{-0.00034}{0.00017} \approx -2 \]For the upper limit (0.23034 mm):\[ Z_{\text{upper}} = \frac{0.23034 - 0.23}{0.00017} = \frac{0.00034}{0.00017} \approx 2 \]

Step 4: Calculate Probability for Y

The Z-scores found are \(Z_{\text{lower}} \approx -2\) and \(Z_{\text{upper}} \approx 2\). Using the standard normal distribution table, we find the probabilities:- \(P(Z \leq -2) \approx 0.0228\)- \(P(Z \leq 2) \approx 0.9772\)Thus, the probability that \(Y\) is within the specified range is:\[ P(-2 < Z < 2) = P(Z \leq 2) - P(Z \leq -2) = 0.9772 - 0.0228 = 0.9544 \]

Step 5: Combine Probabilities for X and Y

Since \(X\) and \(Y\) are independent (given \(\rho = 0\)), the probability that both \(X\) and \(Y\) meet their respective specifications is the product of their individual probabilities:\[ P(X \text{ meets spec}) \times P(Y \text{ meets spec}) = 0.8664 \times 0.9544 = 0.8268 \]

Step 6: Conclusion

The probability that a randomly selected lamp will conform to specifications for both ink layers is approximately 0.8268, or 82.68%.

Key Concepts

Normal Distribution

Understanding normal distribution is crucial in predicting and analyzing data, particularly in manufacturing processes like this one involving electroluminescent lamps. The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. This distribution is bell-shaped and characterized by its mean and standard deviation. In the context of this exercise, the thickness of the ink layers follows a normal distribution.
This allows us to make predictions about the likelihood of a particular thickness occurring. The use of normal distribution helps to estimate the probability that the thickness will be within specified limits, a fundamental aspect when ensuring products meet quality standards.

Z-scores

Z-scores are pivotal when working with normal distributions, as they allow us to determine how far away a specific data point is from the mean. By calculating Z-scores, we translate this into a relative value within the standard normal distribution. The formula to compute a Z-score is:

• \[ Z = \frac{X - \mu}{\sigma} \]

Where:

• \(X\) is the data to be standardized,
• \(\mu\) is the mean,
• \(\sigma\) is the standard deviation.

In this exercise, we calculated Z-scores for the lower and upper bounds of the ink thickness to assess adherence to the desired specifications. The conversion of the thickness measurements into Z-scores enables us to use standard normal distribution tables to find probabilities, making comparisons and other statistical computations straightforward.
With Z-scores, you can easily see how many standard deviations a particular measurement is away from the mean, transforming raw data into a more valuable comparative context.

Independent Events

The concept of independent events plays a significant role in calculating the probability that both ink layers of the lamp conform to the specifications. Two events are independent if the outcome of one does not affect the outcome of the other. In this case, the thickness measurements of ink layers \(X\) and \(Y\) are considered independent because the correlation coefficient \(\rho\) is zero.
When dealing with independent events, the probability of both events occurring is simply the product of their individual probabilities. Here, since \(X\) and \(Y\) are independent, their probabilities can be multiplied, streamlining the calculation of the likelihood that a lamp satisfies both thickness requirements. Recognizing and applying the concept of independent events simplifies complex probability calculations in real-world situations.

Standard Deviation

Standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range. This concept is essential in the standardizing process as it helps in understanding the spread and variability of the data.
In the context of our problem with electroluminescent lamps, the standard deviations for the thickness of ink layers \(X\) and \(Y\) are provided as 0.00031 mm and 0.00017 mm, respectively. These values are crucial for calculating Z-scores and subsequently the probabilities that define whether each thickness falls within the required specifications.

• A low standard deviation, as seen here, indicates precise control over the ink thickness, which is vital for meeting quality standards and maintaining consistency in manufacturing.

Understanding standard deviation enables you to grasp how measurements deviate from the average or expected values, providing insights into the reliability and predictability of a process.