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------------------------------------------------ The lengths of lumber a machine cuts are normally distributed with a mean of 97 inches and a standard deviation of 0.5 inch.

(a) What is the probability that a randomly selected board cut by the machine has a length greater than 97.16 inches?

(b) A sample of 43 boards is randomly selected. What is the probability that their mean length is greater than 97.16 inches?

(a) The probability is (Round to four decimal places as needed.)

To calculate this probability, can use the standard normal distribution (z-distribution) since know the mean and standard deviation of the lengths of lumber.

Given:

Mean (μ) = 97 inches

Standard Deviation (σ) = 0.5 inch

Value (x) = 97.16 inches

need to find the z-score of 97.16 inches and then find the area under the standard normal curve to the right of this z-score.

Z-score formula: z = (x - μ) / σ

z = (97.16 - 97) / 0.5 = 0.32

Using a z-table or calculator, can find the cumulative probability for z = 0.32. The cumulative probability gives us the area under the curve to the left of the z-score. To find the area to the right (greater than 97.16 inches), subtract the cumulative probability from 1.

Probability = 1 - Cumulative Probability(z = 0.32)

Now, calculating this probability:

Probability = 1 - 0.6255 (approximate cumulative probability from the z-table)

Probability ≈ 0.3745

(b) Probability that the mean length of a sample of 43 boards is greater than 97.16 inches:

When dealing with sample means, use the central limit theorem. The distribution of sample means approaches a normal distribution as the sample size increases, even if the original distribution is not normal.

Given:

Sample Size (n) = 43

Sample Mean = 97.16 inches

Population Mean (μ) = 97 inches

Population Standard Deviation (σ) = 0.5 inch

Standard Error (SE) = σ / √n

Calculation:

SE = 0.5 / √43 ≈ 0.0766

Now, need to find the z-score for the sample mean using the formula:

Z-score formula: z = (Sample mean - μ) / SE

z = (97.16 - 97) / 0.0766 ≈ 1.984

Using a z-table or calculator, can find the cumulative probability for z ≈ 1.984. This cumulative probability gives us the area under the standard normal curve to the left of the z-score. To find the area to the right (greater than 97.16 inches), subtract the cumulative probability from 1.

Probability = 1 - Cumulative Probability(z ≈ 1.984)

Now, calculating this probability:

Probability ≈ 1 - 0.9757 (approximate cumulative probability from the z-table)

Probability ≈ 0.0243

(a) The probability that a randomly selected board cut by the machine has a length greater than 97.16 inches is approximately 0.3745.

(b) The probability that the mean length of a sample of 43 boards is greater than 97.16 inches is approximately 0.0243.

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