# The commuting time for all employees of a company has a normal distribution with a mean of 44.6 minutes and a standard deviation of 11.2 minutes. What is the probability that the mean computing time by a sample of 12 employees is between 43.26 and 49.35?

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To find the probability that the mean commuting time by a sample of 12 employees is between 43.26 and 49.35 minutes, you would use the Central Limit Theorem and the properties of the normal distribution. First, you need to find the standard error of the mean, which is calculated by dividing the standard deviation by the square root of the sample size.

Standard Error of the Mean = σ / √n

σ = 11.2 (standard deviation) n = 12 (sample size)

Standard Error of the Mean = 11.2 / √12 ≈ 3.232

Next, you convert the given values of 43.26 and 49.35 to z-scores using the formula:

z = (x - μ) / σ

Where: x = the given value μ = the mean (44.6) σ = the standard deviation (11.2)

For 43.26: z1 = (43.26 - 44.6) / 3.232 ≈ -0.414

For 49.35: z2 = (49.35 - 44.6) / 3.232 ≈ 1.471

Then, you look up the corresponding probabilities in the standard normal distribution table for these z-scores. Once you have those probabilities, you subtract the probability corresponding to the lower z-score from the probability corresponding to the higher z-score to find the probability that the mean commuting time falls between 43.26 and 49.35 minutes.

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