Probability Calculation for Metal Strip Lengths Using Normal Distribution

When evaluating the length distribution of metal strips produced by a machine, it is crucial to understand the principles of normal distribution and z-scores. This article delves into the probability of a randomly selected strip having a length shorter than 165 meters when the lengths are normally distributed with a mean of 150 meters and a standard deviation of 10 meters.

Understanding Normal Distribution and Z-Scores

In statistics, a normal distribution is a continuous 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. The z-score is a measure of how many standard deviations an element is from the mean. The formula for calculating the z-score is:

z (X - μ) / σ

where X is the value for which we want to find the probability, μ is the mean, and σ is the standard deviation.

Case Study: Metal Strip Lengths

Given:

μ (mean) 150 meters σ (standard deviation) 10 meters X (length) 165 meters

To find the probability that a randomly selected strip is shorter than 165 meters:

Step 1: Calculate the Z-Score

The z-score for 165 meters is calculated as:

z (165 - 150) / 10 1.5

Using the standard normal distribution table (or a calculator), the cumulative probability for z 1.5 is approximately 0.9332. This means that about 93.32% of the data falls below this z-score.

Calculating Z-Score with Another Example

Consider another scenario where the mean length of the metal strips is 250 meters and the standard deviation is 10 meters. Now, we want to find the probability that a randomly selected strip is shorter than 230 meters.

Step 1: Calculate the Z-Score

The z-score for 230 meters is calculated as:

z (230 - 250) / 10 -2

Using the standard normal distribution table, the cumulative probability for z -2 is 0.0228. This indicates that there is a small probability (2.28%) that a randomly selected strip is shorter than 230 meters.

Using Online Tools for Probability Calculations

To simplify these calculations, you can use online tools that provide accurate probabilities for normal distribution calculations. These tools can be invaluable for statistical analyses in various fields, including materials science and engineering.

Conclusion

The probability that the length of a randomly selected strip is shorter than 165 meters is approximately 0.9332 or 93.32%. This understanding is essential for quality control and production logistics in industries where materials with specific dimensions are crucial.

Additional Resources for Probability Calculations

For more information on normal distribution and z-scores, consider exploring resources from academic websites, online calculators, and statistical textbooks.

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