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Snails!, Inc., a manufacturer of fasteners, makes a 12d common nail with a mean diameter of 0.14 inches and a standard deviation of 0.012 inches. If a random sample of 32 of these nails is taken, what is the probability that its mean will be greater than 0.143 inches? (Round your answer to four decimal places.)

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Final answer:

The mean and standard deviation for the sample are 0.14 inches and 0.0021238 inches, respectively. The probability that the mean of the sample will be greater than 0.143 inches is approximately 0.0798.

Step-by-step explanation:

In order to find the mean and standard deviation for the sample, we can use the formulas for the mean and standard deviation of a sample mean. The mean of the sample is the same as the mean of the population, which is 0.14 inches. The standard deviation of the sample mean is equal to the standard deviation of the population divided by the square root of the sample size. Therefore, the standard deviation of the sample mean is 0.012 inches divided by the square root of 32, which is approximately 0.0021238 inches.

To find the probability that the mean of the sample will be greater than 0.143 inches, we can use the standard normal distribution. We can standardize the value of 0.143 inches by subtracting the mean of the sample from it and dividing by the standard deviation of the sample mean. This gives us a z-score of (0.143 - 0.14) / 0.0021238, which is approximately 1.408. We can then use a standard normal distribution table or a calculator to find the probability that a standard normal random variable is greater than 1.408. The probability is approximately 0.0798.

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