A sample of n = 49 is drawn at random from a population with a μ = 78 and a σ = 14. What is the probability that the sample mean will be 82 or higher?
A) .3859
B) .9772
C) .0228
D) .6141

Question

Quizplus · Accepted Answer

To solve this problem, we need to use the formula for the sampling distribution of the mean, which tells us that the mean of the distribution of the sample means is equal to the population mean (μ) and the standard deviation of the distribution of the sample means (also known as the standard error) is equal to the population standard deviation (σ) divided by the square root of the sample size (n). 
In this case, the mean of the distribution of the sample means is still 78, but the standard deviation is 14/√49 = 2. Therefore, we want to find the probability that a randomly selected sample of 49 will have a mean of 82 or higher. 
To do this, we need to standardize the sample mean using the formula z = (x̄ - μ) / (σ / √n). 
Thus, z = (82 - 78) / (2) = 2, which corresponds to an area under the standard normal distribution curve of approximately .9772. 
Therefore, the probability that the sample mean will be 82 or higher is approximately .0228 (1 - .9772).

A Sample of N = 49 Is Drawn at Random