1. In a lot of n components, 30% are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective,….

## Find the relative uncertainty in the estimated proportion.

1. The Darcy–Weisbach equation states that the power-generating capacity in a hydroelectric system that is lost due to head loss is given by P = ηγQH, where η is the efficiency of the turbine, γ is the specific gravity of water, Q is the flow rate, and H is the head loss. Assume that η = 0.85 ± 0.02, H = 3.71 ± 0.10 m, Q = 60 ± 1m3 /s, and γ = 9800 N/m3 with negligible uncertainty.

a. Estimate the power loss (the units will be in watts), and find the uncertainty in the estimate.

b. Find the relative uncertainty in the estimated power loss.

c. Which would provide the greatest reduction in the uncertainty in P: reducing the uncertainty in η to 0.01, reducing the uncertainty in H to 0.05, or reducing the uncertainty in Q to 0.5?

2. Let A and B represent two variants (alleles) of the DNA at a certain locus on the genome. Let p represent the proportion of alleles in a population that are of type A, and let q represent the proportion of alleles that are of type B. The Hardy–Weinberg equilibrium principle states that the proportion PAB of organisms that are of type AB is equal to pq. In a population survey of a particular species, the proportion of alleles of type A is estimated to be 0.360 ± 0.048 and the proportion of alleles of type B is independently estimated to be 0.250 ± 0.043.

a. Estimate the proportion of organisms that are of type AB, and find the uncertainty in the estimate.

b. Find the relative uncertainty in the estimated proportion.

c. Which would provide a greater reduction in the uncertainty in the proportion: reducing the uncertainty in the type A proportion to 0.02 or reducing the uncertainty in the type B proportion to 0.02?