Statistical Hypothesis Testing: Interpreting a Z-Test Curve, Exams of Probability and Statistics

Download Statistical Hypothesis Testing: Interpreting a Z-Test Curve and more Exams Probability and Statistics in PDF only on Docsity! 1. Here is a plot of P(reject H0 | mu) as mu varies between 25 and 31. The boundary between H0 and H1 is mu0 = 30. H1 is where P(reject H0 | mu) is large ->|<- H0 a. 1pt Give a fairly accurate numerical value for alpha. Illustrate what you are doing in the plot. ANS. Height of curve at boundary = alpha ~ 0.12 (just be close). b. 1pt Give a fairly accurate numerical value for the power at mu1 = 28. Illustrate what you are doing in the curve. ANS. Height of curve at mu1 = 28 (which is in H1) = power ~ 0.66 (just be close). c. 1pt Identify the null and alternative hypotheses. ANS. H0 = {mu at least 30} (where the curve is low). H1 = {mu < 30} (the boundary is always put in H0). d. 1pt In the above plot, overlay another curve representing P(reject H0 | mu) for a BETTER test of these hypotheses with the SAME ALPHA. ANS. Higher everywhere on H1 and lower everywhere on H0 except at the boundary 30 where it is the same height. It is then suggestive of a better curve for the same alpha. 2. The plot below is P(reject H0 | mu) for mu between 25 and 31. H1 ->|<- H1 a. 1pt Give a fairly accurate numerical value for alpha. Illustrate what you are doing in the curve. ANS. Height of curve at boundary 28.5 = alpha ~ 0.1 (just be close). Draw lines for this in the picture! b. 1pt Give a fairly accurate numerical value for BETA (not power) at mu1 = 27.6. Illustrate what you are doing. ANS. Height of curve at boundary = power ~ 0.6 (just be close). So BETA, the probability of failing to reject H0 when mu is 27.6 = 1 – 0.6 = 0.4 (just be close). Draw lines for this in the picture! c. 1pt Identify the null and alternative hypotheses. ANS. H0 = {mu = 28.5} (where the curve is low). H1 = {mu is not equal to 28.5} (where the curve is high). d. 1pt Overlay on the above plot a curve representing a BETTER test for the SAME ALPHA. ANS. The better curve is everywhere larger than the one given except that on mu = 28.5 it is 0.1 (alpha is the same).