Answer :
(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.
(a) Probability that a randomly selected board is greater than 97.16 inches:
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.
To learn more about probability, visit
https://brainly.com/question/23417919
#SPJ11
