A population has a mean of 70 and a standard deviation of 8. If samples of size N = 16 are randomly drawn from this population, then what proportion of sample means will fall between 66 to 74?
A).6826
B).3413
C).5000
D).9544

Question

Quizplus · Accepted Answer

We will use the Central Limit Theorem here to approximate the sampling distribution with a normal distribution. 
The mean of the sampling distribution will be the same as the population mean, which is 70. 
The standard deviation of the sampling distribution will be the population standard deviation divided by the square root of the sample size, which is 8/sqrt(16) = 2. 
To find the proportion of sample means that fall between 66 and 74, we need to standardize these values using the sampling distribution:

Z1 = (66 - 70)/2 = -2 
Z2 = (74 - 70)/2 = 2

Using a standard normal table or calculator, we can find that the proportion of values between -2 and 2 is approximately 0.9544. Therefore, approximately 95.44% of sample means will fall between 66 and 74. 
So, the answer is D.

A Population Has a Mean of 70 and a Standard