6.2 The criminal justice analogy. This problem was done on the white board in lab during week 8. Briefly, the null hypothesis is analogous to the presumption of innocence. A jury can falsely reject the null hypothesis by declaring an innocent defendant guilty. This is analogous to a type I error. A jury can also falsely retain the null hypothesis by declaring a criminal as "not guilty." This is analogous to a type II error. 

6.3 Patient satisfaction. Done in lab. Also see online key for odd-numbered problems.

6.4 Patient satisfaction survey. ~ N(5.12, 0.125). When = 4.70, z = (4.70 - 5.12) / 0.125 = -3.36. This result lies out in the left tail of the curve. It would be surprising to see this extreme result if the null hypothesis were true.

6.5 Done in lab on the white board. Also see key for odd-numbered problems.

6.6 Lithium. Done in some labs. H₀: � = 1.3  vs. H₁: � 1.3  Notice that will be looking for evidence against the claim that patients are well-maintained in their dosage.

6.7 Done in some lab. Also see key for odd-numbered problems.

6.8 Hemoglobin levels. H₀: � = 12 versus H₁: � 12. SEM = 1.6 / 35 = 0.270. z_stat = (11.2 - 12) / 0.270 = -2.96. Pr(Z < .-2.96) = .0015. Double this, since the alternative hypothesis is two-sided:  p = .0030. The evidence is highly significant against the claim of H₀ (reject H₀).

Comment: This problem is similar to the �anemia� illustrative example presented in class week 8, except the data is different.

6.9 We will try to go over this one in class on 11/1. Also see key for odd-numbered problems. 

6.10 Salary of hospital administrators.  
(A) H₀: � = 85100 versus H₁: �  85100
(B) ~ N(85100, SEM), where SEM = 10000 / 20 = 2236.1.  The sketch should be centered on � = 85000. Landmarks on the curve with landmarks starting at 2 standard errors below the mean at 80,628, 82,864, 85,100, 87,336, 89,572. 
(C)  z_stat = (80900 - 85100) /   2236.1 = -1.88.
(D) The z distribution is drawn with the region to the left of -1.88 shaded. Pr(Z < -1.88) = .0301. The two-sided p value = 2 � .0301 = .0602.
(E) Four explanations for the current findings.
1) The sample is moderate in size and the low sample average could still be due to chance (p = .0602).
2) Perhaps the sample was not random and the investigator selected lower paid female executives on purpose in order to make a point (bad sampling, selection bias).
3) An extraneous factor (lurking variable, confounder) may explain the difference. For example, perhaps male executives are older on the average and older, more experienced executives make more than younger less experienced executives.
4) Comparable women executives actually do make less on average.

6.12 Diet and bowel cancer. The observed difference in polyp re-occurrence between the two groups was so small that it might have occurred purely due to chance, even if we assume diet had no effect..

6.13 See key for odd-numbered problems.