Final answer:
The probability of different sample means can be calculated using the z-score formula and a standard normal distribution table or calculator.
Step-by-step explanation:
The sampling distribution of the sample mean can be described as approximately normal, centered around the population mean of 50. The standard deviation of the sampling distribution, also known as the standard error, can be calculated by dividing the population standard deviation by the square root of the sample size. In this case, the standard error is 10/√49 = 10/7. Calculate the z-scores for each of the given sample means using the formula z = (x - μ) / σ, where x is the sample mean, μ is the population mean, and σ is the standard error. Then, use a standard normal distribution table or a calculator to find the corresponding probabilities.
a. To find the probability that the sample mean is greater than 50, calculate the z-score for a sample mean of 50 using the formula. The probability can be found by subtracting the area to the left of the z-score from 1. P(x > 50) = 1 - P(z < z-score).
b. To find the probability that the sample mean is less than 52, calculate the z-score for a sample mean of 52 using the formula. The probability can be found by finding the area to the left of the z-score. P(x < 52) = P(z < z-score).
c. To find the probability that the sample mean is less than 47, calculate the z-score for a sample mean of 47 using the formula. The probability can be found by finding the area to the left of the z-score. P(x < 47) = P(z < z-score).
d. To find the probability that the sample mean is between 45.5 and 52.5, calculate the z-scores for sample means of 45.5 and 52.5 using the formula. The probability can be found by subtracting the area to the left of the z-score corresponding to 45.5 from the area to the left of the z-score corresponding to 52.5. P(45.5 < x < 52.5) = P(z1 < z < z2).
e. To find the probability that the sample mean is between 50.8 and 51.8, calculate the z-scores for sample means of 50.8 and 51.8 using the formula. The probability can be found by subtracting the area to the left of the z-score corresponding to 50.8 from the area to the left of the z-score corresponding to 51.8. P(50.8 < x < 51.8) = P(z1 < z < z2).