Problem 3. For i = 1,2, let Xi1,,Xịn consistute a sample of size n; from a population having unknown mean u and known variance ož. For X; = +E Xij, i = 1,2, a biostatistician suggests that the "optimal" estimator of the unknown parameter u is û = w,X1+W2X2, where the weights are chosen so that the following two conditions are simultaneously satisfied: • E[i] = µ, i.e., the estimator is unbiased • Varſi], the variance of the estimator, is minimized. Derive explicit expressions for the particular choices of wi and wz that simultaneously satisfy these two conditions, and then interpret your findings.

A weight-loss program wants to test how well their program is working. The company selects a simple random sample of 51 individual that have been using their program for 15 months. For each individual person, the company records the individual's weight when they started the program 15 months ago as an x-value. The subject's current weight is recorded as a y-value. Therefore, a data point such as (205, 190) would be for a specific person and it would indicate that the individual started the program weighing 205 pounds and currently weighs 190 pounds. In other words, they lost 15 pounds. When the company performed a regression analysis, they found a correlation coefficient of r = 0.707. This clearly shows there is strong correlation, which got the company excited. However, when they showed their data to a statistics professor, the professor pointed out that correlation was not the right tool to show that their program was effective. Correlation will NOT show whether or not there is…

8) Consider the data (1,1), (3,9), (4,16), If f[1,3] = 4, f[3,4] = z, f[1,3,4] 1. find z %3D

Problem 1. Consider a battery with a completely unknown voltage (Po = c0). Two independent measurements of the voltage are taken to estimate the voltage, the first with a variance of 4, and the second with a variance of 1. a. Write the weighted least squares voltage estimate in terms of the two measurements z1 and 2.