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HW 4 -- THE NORMAL LOADER 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -3.0 -2.0 -1.0 0.0 1.0 2.0 Z X 3.0 First Name Last Name Total No. of Letters A cement manufacturing company has a machine that drops cement powder mix into 2,000-pound-capacity containers that are positioned beneath it on rail cars. The machine has a dial marked "μ" to control the average (mean) amount of mix dispensed by the machine into each container. The current setting is 2,000 pounds. Like most machines, however, this loader has some variation in its performance. The amounts dispensed per container are normally distributed with a standard deviation of 25 pounds plus the number of letters in your official BU first and last names combined. The instructor’s loader, for example, would have a standard deviation of 25+13=38 pounds. (Many real machines actually have normally-distributed performance patterns.) The amount that enters each container is determined by weighing it after filling. The amount is stamped on the side of the container and the price of the filled container is set accordingly. Thus, underfilled containers are not a problem. But if the loader dispenses more than 2,000 pounds, the excess spills over the sides and is not recoverable. 1. Under current conditions, what percent (four significant figures) of the containers are overfilled, resulting in loss of product? ___________% 2. If management wants the percentage of overfilled containers to be only 2.20000% (0.0220000), to what setting (SIX significant figures) should the dial be moved? ____________lbs 3. Suppose the dial is stuck at 2,000 and cannot be moved. What size container (SIX significant figures) should be used to achieve the 2.2% objective mentioned above? ____________lbs 4. Suppose the container supplier's largest container is 2,030 pounds. If these containers are used, what percentage (four significant figures) of them will be overfilled? ____________% 5. Now the company is using the 2,030-pound containers, but the 2.2% objective is still not being met. The machine maintenance person suggests "tuning up" the machine to make it more consistent (reducing its standard deviation). To what value (four significant figures) must the standard deviation be reduced to achieve the 2.2% objective with the 2,030-pound containers? ____________lbs On the next page, in each numbered section, enter the three known items of information on the appropriate blanks, circle the unknown, then solve for the unknown (show computation in box). In the normal-distribution diagram for each part, add SIX labels: x, μ, z, σ, area, area shading. Turn in both pages. Due at the time of Quiz 4. z = (X - μ) / σ X=μ+zσ μ=X-zσ σ = (X - μ) / z 1. z = __________ 0 .4 5 0 .4 0 .3 5 x = __________ 0 .3 0 .2 5 0 .2 μ = __________ 0 .1 5 0 .1 0 .0 5 σ = __________ 0 - 3 .0 - 2 .0 - 1 .0 0 .0 1 .0 2 .0 Z 3 .0 X 0 .4 5 2. z = __________ 0 .4 0 .3 5 x = __________ 0 .3 0 .2 5 0 .2 μ = __________ 0 .1 5 0 .1 σ = __________ 0 .0 5 0 - 3 .0 - 2 .0 - 1 .0 0 .0 1 .0 2 .0 Z 3 .0 X 0 .4 5 3. z = __________ 0 .4 0 .3 5 x = __________ 0 .3 0 .2 5 0 .2 μ = __________ 0 .1 5 0 .1 σ = __________ 0 .0 5 0 - 3 .0 - 2 .0 - 1 .0 0 .0 1 .0 2 .0 Z 3 .0 X 0 .4 5 4. z = __________ 0 .4 0 .3 5 x = __________ 0 .3 0 .2 5 0 .2 μ = __________ 0 .1 5 0 .1 σ = __________ 0 .0 5 0 - 3 .0 - 2 .0 - 1 .0 0 .0 1 .0 2 .0 Z 3 .0 X 0 .4 5 5. z = __________ 0 .4 0 .3 5 0 .3 x = __________ 0 .2 5 0 .2 μ = __________ 0 .1 5 0 .1 0 .0 5 σ = __________ 0 - 3 .0 - 2 .0 - 1 .0 0 .0 1 .0 2 .0 Z X 3 .0