Main Answer:
(a) The mean of the xbar sampling distribution is 40.
(b) The standard deviation of the xbar sampling distribution is 0.625.
Step-by-step explanation:
In statistics, the mean of the xbar sampling distribution (a) is equal to the population mean, which is 40 in this case. This is a fundamental concept that underscores the idea that, on average, sample means drawn from the same population tend to converge around the population mean.
The standard deviation of the xbar sampling distribution (b) is calculated by dividing the population standard deviation by the square root of the sample size. For this scenario, the standard deviation is 5, and the square root of the sample size (64) is 8. Therefore, 5 divided by 8 equals 0.625, representing the variability or spread of the sample means around the population mean.
In essence, as the sample size increases, the standard deviation of the xbar sampling distribution decreases, indicating a greater precision in estimating the population mean. Conversely, with a smaller sample size, there is more variability in sample means, leading to a higher standard deviation.
Understanding these statistical parameters is crucial for making inferences about a population based on a sample. The mean and standard deviation of the xbar sampling distribution provide insights into the expected value and variability of sample means, forming the foundation for statistical analyses.