1. In a certain type of automobile engine, the cylinder head is fastened to the block by 10 bolts, each of which should be torqued to 60 N · m. Assume that the torques of the bolts are independent.
a. If each bolt is torqued correctly with probability 0.99, what is the probability that all the bolts on a cylinder head are torqued correctly?
b. The goal is for 95% of the engines to have all their bolts torqued correctly. What must be the probability that a bolt is torqued correctly in order to reach this goal?
2. A snowboard manufacturer has three plants, one in the eastern United States, one in the western United States, and one in Canada. Production records show that the U.S. plants each produced 10.000 snowboards last month, while the Canadian plant produced 8000 boards. Of all the boards manufactured in Canada last month. 4% had a defect that caused the boards to de-laminate prematurely. Records kept at the U.S. plants show that 3% of the boards manufactured in the eastern United States and 6% of the boards manufactured in the western United States had this defect as well.
a. What proportion of the boards manufactured last month were defective?
b. What is the probability that a snowboard is defective and was manufactured in Canada?
c. Given that a snowboard is defective, what is the probability that it was manufactured in the United States?
3. The Darey-Weisbach equation states that the power-generating capacity in a hydroelectric system that is lost due to head loss is given by P = ηγ QH, 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 ± 1 m3/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?
4. The intake valve clearances on new engines of a certain type are normally distributed with mean 200 Am and standard deviation 10 μm.
a. What is the probability that the clearance is greater than 215 μm?
b. What is the probability that the clearance is between 180 and 205 μm?
c. An engine has six intake valves. What is the prob-ability that exactly two of them have clearances greater than 215 μm?
5. Polychlorinated biphenyls (PCBs) arc a group of synthetic oil-like chemicals that were at one time widc11 used as insulation in electrical equipment and were discharged into rivers. They were discovered to be a health hazard and were banned in the 1970s. Since then, much effort has gone into monitoring PCB concentrations in waterways. Assume that water samples are being drawn from a waterway in order to estimate the PCB concentration.
a. Suppose that a random sample of size SO has a sample mean of 1.69 ppb and a sample standard deviation of 0.25 ppb. Find a 95% confidence interval for the PCB concentration.
b. Estimate the sample size needed so that a 95% confidence interval will specify the population mean to within ±10.02 ppb.
6. Suppose you have purchased a filling machine for candy bags that is supposed to fill each bag with 16 oz of candy. Assume that the weights of filled bags are approximately normally distributed. A random sample of 10 bags yields the following data (in oz):
15.87 16.02 15.78 15.83 15.69 15.81 16.04 15.81 15.92 16.10
On the basis of these data, can you conclude that the mean fill weight is actually less than 16 oz?
a. State the appropriate null and alternate hypotheses.
b. Compute the value of the test statistic.
c. Find the P-value and state your conclusion.
7. The conversion of cyclobutane (C4H8) to ethylene (C2H4) is a first-outer reaction. This means that the concentration of cyclobutane at time t is given by In C = In C0 - kt, where C is the concentration at lime t, C0 is the initial concentration, t is the time since the reaction started, and k is the rate constant. Assume that C0 = 0.2 mol/L with negligible uncertainty. After 300 seconds at a constant temperature, the concentration is measured to be C = 0.174±0.005 mol/L. Assume that time can be measured with negligible uncertainty.
a. Estimate the rate constant k, and find the uncertainty in the estimate. The units of k will be s-1.
b. Find the relative uncertainty in k.
c. The half-life t1/2 of the reaction is the time it take, rot the concentration to be reduced to one-half its initial value. The halt-life is related to the rate constant by t1/2 = (In 2)/k. Using the result found in part (a), find the uncertainty in the half-life.
d. Find the relative uncertainty in the half-life.