Sampling Distribution Homework Answers Question Set #1 1. The observations are random numbers between 50-100 and have an equal likelihood of occurrence. There is no reason to expect that one number would occur more or less frequently than another. 2. Chart 1 graphs the frequency of the random observations. It is relatively flat. Chart 2 graphs the frequency of the average of two random observations. It is steeper in the middle. This result is intuitive because averaging two observations will tend to reduce the number of extreme occurrences. For example, a value of 50 is just as likely as any other value if we select one random variable. However, if we average two variables, you would have to draw the number 50 twice in a given sample. Any other observation would result in an average greater than 50. Thus, the extreme values will occur with less frequency than the values surrounding the mean. 3. Probably not. The max and min values are near the extremes of our sample possibilities. The standard error is also large. This suggests that we will be very likely to retrieve sample means that are not representative of the population mean. Question Set #2 1. In this problem, the grand mean is the equivalent of a sample mean with 2000 observations. The grand mean closely approximates the population mean. The grand means are all approx. the same. 2. They are never precisely zero, but are close in many instances. The average sampling error is also close to zero, as it is simply the population mean minus the grand mean. 3. It measures the dispersion of sample means around the population mean. It will tell you how much the error deviates from 0 on average. 4. It measures the average dispersion of the error from 0. 5. The sampling error is simply the sample mean minus the population mean. This means that the average sampling error is equal to the average sample mean-the population mean. Therefore, the deviations of the sample means and the sample errors from their respective averages are equal. Hence, their standard deviations are equal. 6. Because the observations in chart 3 are normally distributed, whereas the observations in chart 1 are randomly distributed with an equal likelihood of occurrence. 7. Charts 2 and 4 differ again because the data are normally distributed. The values around the mean have a greater chance of occurring. Therefore, the data are more closely concentrated around the mean. Set #3 1. As n increases, the sample mean will be more likely converge to the population mean and reduce the frequency of values further from the mean. This results in a lower standard error. 2. They are all close because in all 3 instances the grand mean is an average of 2000 samples. While not exactly equal to the population mean, they should be close. 3. Yes, it is a close approximation 4. Chart 5 has a larger sample size. As n increases, the sample mean will be more likely converge to the population mean and reduce the frequency of values further from the mean. 5. As n increases, the sample mean will be more likely converge to the population mean and reduce the frequency of values further from the mean. Thus, the max and min sample means will also converge to the population mean as n becomes larger.