A sample of size 56 will be drawn from a population with mean 26 and standard deviation 11. Find the probability that $\bar { x }$ will be less than 28.
A) 0.9131
B) 0.0869
C) 0.9394
D) 0.9406

Question

Quizplus · Accepted Answer

To find the probability that $\bar{x}$ will be less than 28, we use the formula for the standard error of the mean: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. Here, $\sigma = 11$ and $n = 56$, so $\sigma_{\bar{x}} = \frac{11}{\sqrt{56}}$. Then, we calculate the z-score for $\bar{x} = 28$ using $z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$, where $\mu = 26$. After finding the z-score, we look up the corresponding probability in the standard normal distribution table or use a calculator. The calculation leads to a probability of 0.9131 that $\bar{x}$ will be less than 28.

A Sample of Size 56 Will Be Drawn from a Population