Consider the data set (in the given order): 5 7 9 4 5 6 Given that ˆ 1 (k=1) = -
ID: 3124302 • Letter: C
Question
Consider the data set (in the given order):
5 7 9 4 5 6
Given that ˆ1 (k=1) = -1/8, ˆ2 (k=2) = -1/2 and ˆj = 0 (j3), find the approximate estimated standard deviation of autocorrelation of lag-2 (ˆ2). Discuss the reliability of this value.
Was not sure exactly what standard deviation it is looking for. I have ([(5-6)2+(7-6)2+(9-6)2+(4-6)2+(5-6)2+(6-6)2]/(6)) = 1.6329 as the population standard deviation if that helps.
Bonus Question: when you are given a data set of 6 numbers like above, do you consider that a population and use population standard deviation or sample standard deviation?
Explanation / Answer
We are normally interested in knowing the population standard deviation because our population contains all the values we are interested in. Therefore, you would normally calculate the population standard deviation if: (1) you have the entire population or (2) you have a sample of a larger population, but you are only interested in this sample and do not wish to generalize your findings to the population. However, in statistics, we are usually presented with a sample from which we wish to estimate (generalize to) a population, and the standard deviation is no exception to this. Therefore, if all you have is a sample, but you wish to make a statement about the population standard deviation from which the sample is drawn, you need to use the sample standard deviation. Confusion can often arise as to which standard deviation to use due to the name "sample" standard deviation incorrectly being interpreted as meaning the standard deviation of the sample itself and not the estimate of the population standard deviation based on the sample.
S(ˆ2) = ((1+2* ˆ2^2 + 2*ˆ3^2)*(1/6)) = 0.5052 The above is the sample standard deviation estimate for n = 6, thus it is not extremely reliable estimate since n is not sufficiently large. Furthermore, calculated by hand, ˆ3 = ˆ4 = 0.0625