Question

the lengths of lumber a machine cuts are normally distributed with a mean of 103 inches and a standard deviation of 0.6 inch. (a) what is the probability that a randomly selected board cut by the machine has a length greater than 103.26 inches? (b) a sample of 42 boards is randomly selected. what is the probability that their mean length is greater than 103.26 inches? (a) the probability is 0.3336 (round to four decimal places as needed.) (b) the probability is (round to four decimal places as needed.)

Explanation:

Step1: Calculate the standard error

The standard error \( \sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}} \), where \( \sigma = 0.6 \) and \( n = 42 \).
\( \sigma_{\bar{x}}=\frac{0.6}{\sqrt{42}}\approx0.0926 \)

Step2: Calculate the z - score

The z - score formula is \( z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}} \), where \( \bar{x}=103.26 \), \( \mu = 103 \)
\( z=\frac{103.26 - 103}{0.0926}=\frac{0.26}{0.0926}\approx2.81 \)

Step3: Find the probability

We want \( P(\bar{X}>103.26) \), which is equivalent to \( P(Z > 2.81) \)
Since \( P(Z>z)=1 - P(Z\leq z) \), and from the standard normal table \( P(Z\leq2.81)=0.9975 \)
\( P(Z > 2.81)=1 - 0.9975=0.0025 \)

Answer:

\(0.0025\)