Answer :
Final answer:
The probability that a randomly selected board has a length greater than 97.17 inches is 33.72%. The probability that the mean length of a sample of 39 boards is greater than 97.17 inches is 0.39%.
Explanation:
(a) To find the probability that a randomly selected board has a length greater than 97.17 inches, we need to find the area under the normal curve to the right of 97.17.
To do this, we can calculate the z-score using the formula: z = (x - mean) / standard deviation.
Plugging in the values, we find z = (97.17 - 97) / 0.4 = 0.42.
Using a standard normal distribution table or a calculator, we find that the probability of a randomly selected board having a length greater than 97.17 inches is approximately 0.3372, or 33.72%.
(b) To find the probability that the mean length of a sample of 39 boards is greater than 97.17 inches, we need to find the standard error of the mean. The standard error of the mean is calculated as standard deviation / sqrt(sample size).
In this case, the standard error of the mean is 0.4 / sqrt(39) = 0.064.
Using the z-score formula again, we find z = (97.17 - 97) / 0.064 = 2.66.
Referring to a standard normal distribution table or a calculator, we find that the probability of the mean length of a sample of 39 boards being greater than 97.17 inches is approximately 0.0039, or 0.39%.
Learn more about normal distribution here:
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