Statistics for laboratory scientists

Solutions for the homework problems for lecture 10

1. 1. Xbar ~ normal(mean=2, SD = 1.5/sqrt(15) = 0.387). Let Z = (Xbar - 2)/0.387. Then Pr(Xbar > 3.2) = Pr[Z > (3.2-2.0)/0.387)] = Pr(Z > 3.10) = (approx) 1/1000.

   2. Xbar ~ normal(mean=3, SD = 0.387). Let Z = (Xbar - 3)/0.387. Pr(Xbar > 3.2) = Pr[Z > (3.2-3.0)/0.387] = Pr(Z > 0.516) = (approx) 30%.

   3. Xbar ~ normal(mean=2, SD=3/sqrt(15) = 0.775). Let Z = (Xbar - 2)/0.775. Pr(Xbar > 3.2) = Pr[Z > (3.2-2.0)/0.775] = Pr(Z > 1.55) = (approx) 6%.

   4. Xbar ~ normal(mean=3, SD = 0.775). Let Z = (Xbar - 3)/0.775. Pr(Xbar > 3.2) = Pr[Z > (3.2-3.0)/0.775] = Pr(Z > 0.258) = (approx) 40%.

   5. Xbar ~ normal(mean=2, SD = 1.5/sqrt(3) = 0.866). Let Z = (Xbar - 2)/0.866. Then Pr(Xbar > 3.2) = Pr[Z > (3.2-2.0)/0.866)] = Pr(Z > 1.36) = (approx) 8%.

   6. Xbar ~ normal(mean=3, SD = 0.886). Let Z = (Xbar - 3)/0.886. Pr(Xbar > 3.2) = Pr[Z > (3.2-3.0)/0.886] = Pr(Z > 0.231) = (approx) 41%.

   7. Xbar ~ normal(mean=2, SD = 1.5/sqrt(100) = 0.15). Let Z = (Xbar - 2)/0.15. Then Pr(Xbar > 3.2) = Pr[Z > (3.2-2.0)/0.15)] = Pr(Z > 8) = (approx) 0.

   8. Xbar ~ normal(mean=3, SD = 0.15). Let Z = (Xbar - 3)/0.15. Pr(Xbar > 3.2) = Pr[Z > (3.2-3.0)/0.15] = Pr(Z > 1.33) = (approx) 9%.

2. Xbar ~ normal(mean=10, sd=2.5/sqrt(100)=0.25). Let Z = (Xbar-10)/0.25.

   1. Pr(|Xbar - 10| < 0.1) = Pr(|Z| < 0.1/0.25) = Pr(|Z| < 0.4) = (approx) 31%.

   2. Pr(Xbar > 10.25) = Pr(Z > 1) = (approx) 16%.