The processing time for the robogate has a normal distribution with mean 23 sec and standard deviation 2 sec. Find the probability that the next operation of the robogate will take $26.8 \mathrm{sec}$ or less.
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The probability is
Answer
Final Answer: The probability that the next operation of the robogate will take 26.8 seconds or less is approximately \(\boxed{0.9713}\).
Step 1 ：The problem is asking for the probability that the processing time is less than or equal to 26.8 seconds. This is a problem of normal distribution.
Step 2 ：The first step is to standardize the value 26.8 by subtracting the mean and dividing by the standard deviation. This will give us the z-score. The formula for calculating the z-score is \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 3 ：Substituting the given values into the formula, we get \(Z = \frac{26.8 - 23}{2} = 1.9\).
Step 4 ：Then, we can use the standard normal distribution table to find the probability that corresponds to this z-score. The probability corresponding to a z-score of 1.9 is approximately 0.9713.
Step 5 ：Final Answer: The probability that the next operation of the robogate will take 26.8 seconds or less is approximately \(\boxed{0.9713}\).
