Final answer:
a. The central limit theorem states that the sample mean follows a normal distribution when the sample size is large. b. The probability that the sample mean exceeds a specified value can be calculated using the z-score. c. The probability that the sample mean equals the population mean is low but can be calculated under certain conditions. d. The sample size affects the distribution of sample means, making it more normal and reducing the standard deviation.
Step-by-step explanation:
a. Probability that the sample mean follows a normal distribution:
The central limit theorem states that when the sample size is large enough, the distribution of sample means will be approximately normal, regardless of the distribution of the population. This means that the sample mean is more likely to follow a normal distribution when the sample size is large.
b. Probability that the sample mean exceeds a specified value:
To find this probability, you would need the population mean and standard deviation, as well as the sample size. With that information, you can calculate the z-score of the specified value and use the standard normal distribution table to find the probability.
c. Probability that the sample mean equals the population mean:
The probability that the sample mean equals the population mean is very low, especially since sample means tend to have some variation. However, if the sample size is very large, and the population mean and standard deviation are known, the probability can be calculated.
d. How the sample size affects the distribution of sample means:
As the sample size increases, the distribution of sample means becomes more normal. This is because larger sample sizes provide more information about the population, resulting in a more accurate estimate of the population mean. Additionally, the standard deviation of the distribution of sample means decreases as the sample size increases.