To find the probability that the mean from a sample of 64 observations differs from the population mean of 180, we can use the Central Limit Theorem. According to this theorem, the sampling distribution of the sample mean will be normally distributed if the sample size is sufficiently large (which it is in this case).
The mean of the sampling distribution will be equal to the mean of the population, which is 180. The standard deviation of the sample mean (also known as the standard error) can be calculated using the formula:
Standard Error (SE) = σ / √n
Where:
- σ = population standard deviation (24 in this case)
- n = sample size (64)
Plugging in these values, we get:
SE = 24 / √64 = 24 / 8 = 3
Now, we need to determine the probability that the sample mean is within a certain range of the population mean. Let’s say we want to know the probability that the sample mean is between 177 and 183. To find this, we will standardize these values (convert them to z-scores):
Z = (X – μ) / SE
Where:
- X = value we are interested in
- μ = population mean (180)
- SE = standard error (3)
Calculating z-scores for 177 and 183:
Z for 177 = (177 – 180) / 3 = -1
Z for 183 = (183 – 180) / 3 = 1
Now, we can look up these z-scores in the standard normal distribution table or use a calculator to find the probabilities:
The probability corresponding to Z = -1 is approximately 0.1587, and for Z = 1 it is approximately 0.8413. Therefore, the probability that the sample mean falls between 177 and 183 is:
Probability = P(Z < 1) - P(Z < -1) = 0.8413 - 0.1587 = 0.6826
Thus, there is approximately a 68.26% chance that the mean from a sample of 64 observations will fall between 177 and 183.