A sample of size 70 will be drawn from a population with mean 24 and standard deviation 10. Find the probability that $\bar { x }$ will be between 22 and 25 .
A) 0.7521
B) 0.0475
C) 0.7326
D) 0.2005

Question

A sample of size 70 will be drawn from a population with mean 24 and standard deviation 10. Find the probability that  $\bar { x }$  will be between 22 and 25 .
A)  0.7521 
B)  0.0475 
C)  0.7326 
D)  0.2005

Quizplus · Accepted Answer

The probability that $\bar{x}$ will be between 22 and 25 can be found using the Central Limit Theorem, which states that the sampling distribution of the sample mean will be normally distributed (or approximately so) if the sample size is large enough (n > 30 is a common rule of thumb), regardless of the shape of the population distribution. Given a population mean ($\mu$) of 24, a population standard deviation ($\sigma$) of 10, and a sample size (n) of 70, we can calculate the standard error of the mean ($\sigma_{\bar{x}}$) as $\sigma / \sqrt{n}$ = 10 / $\sqrt{70}$ ≈ 1.195. To find the probability that $\bar{x}$ is between 22 and 25, we convert these x values to z-scores using the formula $z = (\bar{x} - \mu) / \sigma_{\bar{x}}$. For $\bar{x} = 22$, $z = (22 - 24) / 1.195 ≈ -1.67$. For $\bar{x} = 25$, $z = (25 - 24) / 1.195 ≈ 0.84$.Using the standard normal distribution table or a calculator, we find the probabilities corresponding to these z-scores and subtract the smaller from the larger to find the probability that $\bar{x}$ falls between 22 and 25. This calculation yields a probability of approximately 0.7521, making option A the correct choice.

A Sample of Size 70 Will Be Drawn from a Population