Question 1. The distribution of the number of employees that new start-ups in K/W involve follows a normal distribution with an unknown mean and a known variance V2 = 1. Suppose you are interested in using a sample with n = 16 observations to test the null hypothesis H0: P ≤ 4, and the alternative hypothesis H1: P > 4. i) Define the decision rule for a test at the 5% significance level. ii) If you obtain a sample and calculate that the sample mean 4.8 employees per firm, what is the outcome of the test? What is the p-value of the test? iii) Now disregard the particular sample in ii). Based upon your decision rule in i), calculate the probability of type II error if the true value of the population mean is a) P = 4.25 b) P = 4.5 Provide a sketch of the true and hypothesized distributions of the sample mean for each case and indicate the power of the test. Question 2. You wish to estimate the difference in savings (in thousands $) of two groups by estimating the model, �௜௝ = � + �௝ + �௜ where the two parameters µ and δj represent the grand mean and the group contrasts between groups j =1,2. You will estimate µ and δj (assuming δ = δ1 = -δ2 ) by projecting an observation vector Y with 3 observations from each group onto the basis for the "model space", ⎩ ⎪ ⎨ ⎪ ⎧ ⎝ ⎜ ⎛ 1 1 1 1 1 1⎠ ⎟ ⎞ , ⎝ ⎜ ⎛ 1 1 1 −1 −1 −1⎠ ⎟ ⎞ ⎭ ⎪ ⎬ ⎪ ⎫ i) Let �መ be the least squares estimator of the contrast parameter δ. Derive �ൣ�መ൧ ��� ���ൣ�መ൧. ii) What is your estimate of δ when the observation vector obtained (negative values representing debt) is � = ⎝ ⎜ ⎛ 3 3 5 −2 3 1 ⎠iii) Using the data in ii), test the hypothesis that δ = 0 against the alternative δ > 0 at the 5% significance level. What do you conclude? Question 3. The population model �௜ = � + �௜ with independent errors �௜ ~ �(0, �ଶ), implies that a random sample of size n can be represented as, � = ൮ �ଵ �ଶ ⋮ �௡ ൲ = � ቌ 1 1 ⋮ 1 ቍ + ൮ �ଵ �ଶ ⋮ �௡ ൲. i) Project the observation vector � = ⎝ ⎜ ⎛ 4.2 8.4 1.4 5.4 3.2⎠ ⎟ ⎞ onto the unit vector �ଵ = ଵ √ହ ⎝ ⎜ ⎛ 1 1 1 1 1⎠ ⎟ ⎞ . What is the projection coefficient, and what is the projection vector? What is your estimate of �? ii) Find the residual vector and illustrate the orthogonal decomposition with a triangle with sides corresponding to observation vector, the projection vector (or fitted vector) and residual vector. What are the lengths of these vectors? iii) To test H0: µ = 0 vs. HA: µ ≠ 0, construct the F-statistic � = (௒∙௎భ)మ ‖௘‖మ ௡ିଵ ൗ = ௡௒തమ ௦మ ~�ଵ,௡ିଵ using your answer in ii). Undertake the test at the 1% significance level. What do you conclude?