Question

Let Yı, Y2, Y3, Y4 represent the values in a simple random sample (SRS) drawn from a population with mean u and standard deviation o. Recall that the "SRS" part means the observations on Y are independent and identically distributed (or "iid"). This in turn means that E[Y1] = E[Y2] = E[Y3] = E[Y4] = u, and Var[Y1] = Var[Y2] = Var[Y3] = Var[Y4] = o2, and Cov(Y1, Y2) = Cov(Y1, Y3) = Cov(Y1, Y4) = Cov(Y2, Y3) = Cov(Y2, Y4) = Cov(Y3, Y4) = 0. (For short, we would often just write this as E[Yi] = u, Var[Yi] = 02, Vi and Cov(Yi, Y;) = 0 Vi #j, where 'V' stands for "for all.") + Show that the sample average Y = 1/4(Y1+Y2+Y3+Y4) is unbiased (remind yourself what the definition of an unbiased estimator is).

Answer 1

The sample average Y = (1/4)(Y1 + Y2 + Y3 + Y4) is unbiased because its expected value is equal to the population mean u.

Step-by-step explanation:

A sample average is said to be unbiased if its expected value is equal to the population mean. To show that the sample average Y = (1/4)(Y1 + Y2 + Y3 + Y4) is unbiased, we need to calculate its expected value.

Since E[Y] = u, we can conclude that the sample average Y is unbiased.