Final answer:
The standard error of the mean is 0.78, and the mean of the sampling distribution equals the population mean, which is 3. The population variance affects the spread of the sampling distribution, and the sample mean approximates the population mean over many samples.
Step-by-step explanation:
The student's query involves finding the mean of the sampling distribution, calculating the standard error of the mean, understanding the impact of the population variance on the sampling distribution, and comparing the sample mean to the population mean.
The standard error of the mean (SE) is given by the formula SE = σ/√n, where σ is the population standard deviation and n is the sample size. In this case, since the variance is 49, σ equals the square root of 49, which is 7. The sample size (n) is 81, so the SE = 7/√81, which simplifies to SE = 7/9. The standard error of the mean, rounded to two decimal places, is therefore 0.78.
The mean (μ) of the sampling distribution is always equal to the population mean, assuming the law of large numbers holds. As given, the population mean is 3, so the sampling distribution mean is also 3.
A higher population variance tends to increase the standard error and therefore the spread of the sampling distribution, making it wider. Conversely, a lower variance tends to result in a smaller standard error and a more concentrated sampling distribution.
The sample mean is an unbiased estimator of the population mean, so over many samples, the average of the sample means would converge to the population mean of 3.