Model Answer
0 min readIntroduction
In quality control and manufacturing processes, understanding the distribution of product characteristics is crucial for ensuring quality and minimizing defects. The normal distribution is a fundamental statistical tool used to model such characteristics, like the weight of parts produced. By knowing the mean and standard deviation, we can calculate the probability of a part falling within a specific weight range. This question requires us to determine the probability that a part’s weight lies between 65 kg and 89 kg, given a normal distribution with a mean of 73 kg and a standard deviation of 8 kg.
Understanding the Problem
We are given that the weights of parts are normally distributed. This means the distribution is symmetrical and bell-shaped, defined by its mean (μ) and standard deviation (σ). Here, μ = 73 kg and σ = 8 kg. We need to find the probability P(65 ≤ X ≤ 89), where X represents the weight of a part.
Calculating Z-Scores
To find this probability, we first need to convert the weights (65 kg and 89 kg) into Z-scores. The Z-score represents the number of standard deviations a particular value is away from the mean. The formula for calculating the Z-score is:
Z = (X - μ) / σ
Calculating Z-score for 65 kg
Z1 = (65 - 73) / 8 = -8 / 8 = -1
Calculating Z-score for 89 kg
Z2 = (89 - 73) / 8 = 16 / 8 = 2
Using the Standard Normal Distribution Table
Now we need to find the probability corresponding to these Z-scores. We are looking for P(-1 ≤ Z ≤ 2). This can be calculated as P(Z ≤ 2) - P(Z ≤ -1). We can use a standard normal distribution table (also known as a Z-table) to find these probabilities.
Finding P(Z ≤ 2)
Looking up Z = 2 in a standard normal distribution table, we find P(Z ≤ 2) ≈ 0.9772.
Finding P(Z ≤ -1)
Looking up Z = -1 in a standard normal distribution table, we find P(Z ≤ -1) ≈ 0.1587.
Calculating the Probability
Therefore, P(-1 ≤ Z ≤ 2) = P(Z ≤ 2) - P(Z ≤ -1) = 0.9772 - 0.1587 = 0.8185
Final Answer
The probability that a part coming out from this process will weigh between 65 kg and 89 kg is approximately 0.8185 or 81.85%.
Conclusion
In conclusion, by utilizing the concept of the normal distribution and calculating Z-scores, we were able to determine the probability of a part’s weight falling within the specified range. This demonstrates the practical application of statistical tools in quality control and process management. Understanding these principles is vital for optimizing production processes and ensuring product consistency. The ability to interpret and apply statistical data is a key skill for effective management decision-making.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.