Final answer:
To find the probability that the sample mean is between the target and the population mean of 0.873, we calculate the standard error of the mean, the z-scores for the target mean and population mean, and use a standard normal distribution table or a calculator to find the associated probabilities. The probability that the sample mean is between the target and the population mean is 0.5.
Step-by-step explanation:
To find the probability that the sample mean is between the target and the population mean of 0.873, we need to calculate the z-scores for both values and their corresponding probabilities.
First, we calculate the standard error of the mean using the formula SE = σ / √n, where σ is the standard deviation of the population and n is the sample size.
SE = 0.004 / √24 = 0.000816
Next, we calculate the z-scores for both the target mean (0.87) and the population mean (0.873) using the formula z = (x - μ) / SE, where x is the mean we want to find the probability for and μ is the population mean.
For the target mean of 0.87, z = (0.87 - 0.873) / 0.000816 = -3.67
For the population mean of 0.873, z = (0.873 - 0.873) / 0.000816 = 0
Using a standard normal distribution table or a calculator, we can find the probability associated with each z-score. The probability for the target mean (z = -3.67) is approximately 0, and the probability for the population mean (z = 0) is 0.5.
To find the probability that the sample mean is between the target and the population mean, we subtract the probability of the target mean (0) from the probability of the population mean (0.5):
Probability = 0.5 - 0 = 0.5