A sample of size 65 will be drawn from a population with mean 22 and standard deviation 15 . Find the probability that $\bar { x }$ will be between 20 and 25 .
A) 0.0537
B) 0.8040
C) 0.7465
D) 0.1423

Question

Quizplus · Accepted Answer

To find the probability that the sample mean $\bar{x}$ will be between 20 and 25, we use the Central Limit Theorem, which states that the sampling distribution of the sample mean will be approximately normally distributed if the sample size is large enough (usually n > 30 is considered sufficient). The mean of the sampling distribution of the sample mean is equal to the population mean ($\mu$), and the standard deviation of the sampling distribution of the sample mean (standard error) is equal to the population standard deviation ($\sigma$) divided by the square root of the sample size (n), which is $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.Given:- Population mean ($\mu$) = 22- Population standard deviation ($\sigma$) = 15- Sample size (n) = 65First, calculate the standard error ($\sigma_{\bar{x}}$):$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{65}} \approx 1.86$Next, find the z-scores for 20 and 25:$$Z_{20} = \frac{20 - \mu}{\sigma_{\bar{x}}} = \frac{20 - 22}{1.86} \approx -1.08$$$$Z_{25} = \frac{25 - \mu}{\sigma_{\bar{x}}} = \frac{25 - 22}{1.86} \approx 1.61$$Using the standard normal distribution table or a calculator, find the probabilities corresponding to these z-scores:- The probability of z being less than -1.08 is approximately 0.1401.- The probability of z being less than 1.61 is approximately 0.9463.The probability that $\bar{x}$ will be between 20 and 25 is the difference between these two probabilities:$$P(20 < \bar{x} < 25) = P(Z < 1.61) - P(Z < -1.08) \approx 0.9463 - 0.1401 = 0.8062$$The closest answer choice to this calculated probability is 0.8040, making option B the correct choice.

A Sample of Size 65 Will Be Drawn from a Population