A sample of size 50 will be drawn from a population with mean 10 and standard deviation 8 . Use the Cumulative Normal Distribution Table if needed.

Part: 0 / 2

Part 1 of 2
(a) Find the probability that $\bar{x}$ will be greater than 8 . Round the final answer to at least four decimal places. The probability that $\bar{x}$ will be greater than 8 is

Step 1 ：The problem is asking for the probability that the sample mean (\(\bar{x}\)) will be greater than 8. This is a problem of normal distribution. We know that the sample mean follows a normal distribution with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.

Step 2 ：We can use the z-score formula to standardize the value of 8 and then use the cumulative distribution function (CDF) of the standard normal distribution to find the probability. The z-score formula is: \(z = (\bar{x} - \mu) / (\sigma / \sqrt{n})\) where \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Step 3 ：After finding the z-score, we can use the CDF to find the probability that z is less than the calculated z-score. Since we want the probability that \(\bar{x}\) is greater than 8, we need to subtract the calculated probability from 1.

Step 4 ：Given that the population mean \(\mu\) is 10, the population standard deviation \(\sigma\) is 8, and the sample size \(n\) is 50, we can substitute these values into the z-score formula to find the z-score.

Step 5 ：Substituting \(\bar{x} = 8\), \(\mu = 10\), \(\sigma = 8\), and \(n = 50\) into the z-score formula, we get \(z = -1.7677669529663689\).

Step 6 ：Using the CDF, we find that the probability corresponding to this z-score is 0.9614500641282292.

Step 7 ：Since we want the probability that \(\bar{x}\) is greater than 8, we subtract the calculated probability from 1 to get the final answer.

Step 8 ：Final Answer: The probability that \(\bar{x}\) will be greater than 8 is \(\boxed{0.9614}\).