The lengths of lumber a machine cuts are normally distributed with a mean of 86 inches and a standard deviation of 0.3 inch.
(a) What is the probability that a randomly selected board cut by the machine has a length greater than 86.13 inches?
(b) A sample of 40 boards is randomly selected. What is the probability that their mean length is greater than 86.13 inches?
(a) The probability is
(Round to four decimal praces as needed.)
Answer
Final Answer: The probability that a randomly selected board cut by the machine has a length greater than 86.13 inches is approximately \(\boxed{0.6676}\).
Step 1 ：The problem is asking for the probability that a randomly selected board cut by the machine has a length greater than 86.13 inches. This is a problem of normal distribution.
Step 2 ：We can use the z-score formula to calculate the z-score, which is \((X - μ) / σ\), where X is the value we are interested in, μ is the mean, and σ is the standard deviation.
Step 3 ：Substitute the given values into the formula: mean = 86, standard deviation = 0.3, and X = 86.13.
Step 4 ：Calculate the z-score: \(z = (86.13 - 86) / 0.3 = 0.4333333333333182\).
Step 5 ：After we get the z-score, we can use the standard normal distribution table or a function to get the probability. However, the table or function usually gives the probability that a value is less than X, so we need to subtract the result from 1 to get the probability that a value is greater than X.
Step 6 ：Calculate the probability: \(P = 1 - 0.3323863126266806 = 0.6676\).
Step 7 ：Final Answer: The probability that a randomly selected board cut by the machine has a length greater than 86.13 inches is approximately \(\boxed{0.6676}\).
