Final answer:
To calculate the probability of not detecting the change, determine the standard deviation, calculate the z-score, and use a standard normal distribution table or calculator to find the probability.
Step-by-step explanation:
To calculate the probability of not detecting the change as soon as the first sample taken after the change, we need to calculate the probability of observing a sample mean within the control limits given the new average of non-conforming products.
The control limits are based on the 3-sigma rule, which means that the control limits are set at 3 standard deviations above and below the mean. So, if the new average of non-conforming products is 5% - 2% = 3%, we can calculate the standard deviation using the formula: σ = √(p * (1-p) / n), where p is the average of non-conforming products and n is the sample size.
Once we have the standard deviation, we can calculate the z-score for the new average using the formula: z = (x - μ) / σ, where x is the new average, μ is the previous average, and σ is the standard deviation. With the z-score, we can calculate the probability using a standard normal distribution table or a calculator.