statistics
posted by sophia on .
Given a distribution with mean X and standard deviation S, answer the following.
4.1 Determine the mean and standard deviation if constant K is added to each observation
4.2 Determine the mean and standard deviation if constant K is multiplied by each observation
4.3 Assuming no knowledge about the distribution, estimate the percentage of observations that lie within two (2) standard deviations of the mean; explain your approach
4.4 Assuming the distribution is normally distributed, estimate the percentage of observations that lie within two (2) standard deviations of the mean; explain your approach

I'll explain 2 of these questions; see if you can figure out the other two.
4.1 The mean will be changed and the standard deviation will remain the same. There will be no change in standard deviation since each score and also the mean increase by constant K.
4.3 You can use Chebyshev's theorem for any distribution. This theorem says that within 2 standard deviations of the mean, you will always find at least 75% of the data.
There is a formula to show this:
1  (1/k^2)
If k = 2 (representing 2 standard deviations), then 1  (1/2^2) = 1  1/4 = 3/4 = .75 or 75%.
For 4.2: Calculate mean and standard deviation using an example data set, then multiply each observation in the data set by a constant. See what happens to the mean and standard deviation.
For 4.4: Use the 689599.7 rule which is also called the Empirical Rule.
I hope this will help.