A population has a mean of 300 and a standard deviation of 50. Suppose a sample of size 125 is selected and is used to estimate .

What is the probability that the sample mean will be within +/- 3 of the population mean (to 4 decimals)?

To find the probability that the sample mean will be within +/- 3 of the population mean, we can use the Central Limit Theorem.

The Central Limit Theorem states that for a large sample size, the distribution of sample means will be approximately normally distributed, regardless of the shape of the underlying population. Additionally, the mean of the sample means will be equal to the population mean, and the standard deviation of the sample means (also known as the standard error) will be equal to the population standard deviation divided by the square root of the sample size.

In this case, the population mean is 300, the population standard deviation is 50, and the sample size is 125.

First, we need to calculate the standard error of the sample means. Using the formula:

Standard Error = Population Standard Deviation / √Sample Size

Standard Error = 50 / √125

Standard Error ≈ 4.47 (rounded to 2 decimal places)

Next, we need to calculate the z-scores for the lower and upper boundaries of +/- 3.

z-score for lower boundary = (Lower Mean Bound - Population Mean) / Standard Error
z-score for lower boundary = (300 - 300) / 4.47
z-score for lower boundary = 0

z-score for upper boundary = (Upper Mean Bound - Population Mean) / Standard Error
z-score for upper boundary = (300 + 3 - 300) / 4.47
z-score for upper boundary = 0.67

Using a z-table or a calculator, we can find the probability associated with these z-scores.

The probability associated with a z-score of 0 is 0.5000.

The probability associated with a z-score of 0.67 is 0.7486.

To find the probability that the sample mean will be within +/- 3 of the population mean, we subtract the probability associated with the lower boundary from the probability associated with the upper boundary:

Probability = 0.7486 - 0.5000

Probability = 0.2486

So, the probability that the sample mean will be within +/- 3 of the population mean is approximately 0.2486 (rounded to 4 decimal places).