Final answer:
The probability that a sample mean from a normally distributed population falls between 139.6 and 139.9 with the given parameters is 0.0014. This is found by calculating the z-scores for the sample means and subtracting the corresponding probabilities.
Step-by-step explanation:
To calculate the probability that a sample mean is between 139.6 and 139.9 for a normally distributed population with μ = 146.6 and σ = 30.1, and with a sample size of n = 136, we apply the Central Limit Theorem. This theorem tells us that the sampling distribution of the sample mean will be normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size (σ/√n).
The standard deviation of the sample means, also known as the standard error, is calculated as σ/√n = 30.1/√136 = 2.5775. Next, we find the z-scores for 139.6 and 139.9 using the formula Z = (X - μ)/(σ/√n). For 139.6, Z = (139.6 - 146.6)/2.5775 = -2.7179, and for 139.9, Z = (139.9 - 146.6)/2.5775 = -2.5966.
Using a z-table or calculator, we find the probability corresponding to these z-scores. The probability for Z = -2.7179 is approximately 0.0033, and for Z = -2.5966 is approximately 0.0047. The probability of the sample mean being between 139.6 and 139.9, P(139.6 < M < 139.9), is the difference between these two probabilities, which is 0.0047 - 0.0033 = 0.0014.
Therefore, P(139.6 < M < 139.9) = 0.0014, accurate to four decimal places.