Question

For a sample of n=40​, find the probability of a sample mean

being less than 12,748 or greater than 12,751 when μ=12,748 and
σ=1.7

Answer 1

To find the probability that the sample mean is between 42 and 50, we can use the Central Limit Theorem. The probability of the sample mean being between 42 and 50 is 0.6806.

Step-by-step explanation:

To find the probability that the sample mean is between 42 and 50, we can use the Central Limit Theorem. According to the Central Limit Theorem, if the sample size is large enough (n ≥ 30), the distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.

First, we need to find the z-scores of the upper and lower limits of the sample mean. The z-score formula is z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

For the lower limit, z = (42 - 45) / (8 / √30) = -0.836, and for the upper limit, z = (50 - 45) / (8 / √30) = 1.19.

Next, we use a z-table or calculator to find the area under the standard normal distribution curve corresponding to these z-scores. The probability of the sample mean being between 42 and 50 is the difference between the two areas: P(42 < x < 50) = P(z < 1.19) - P(z < -0.836).

Using a z-table or calculator, we find that P(z < 1.19) = 0.8821 and P(z < -0.836) = 0.2015. Therefore, P(42 < x < 50) = 0.8821 - 0.2015 = 0.6806.