2005%20fall - Math 218 Final Examination Fall 2005...

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Math 218 Final Examination: Fall 2005 Directions. On this examination, you may use a calculator and one 8-1/2 by 11-inch sheet of handwritten notes (both sides may be written on). No books or other notes are permitted. When an answer box is provided, copy your answer into that box. Numerical answers should be evaluated to be either decimals or fractions. Numerical answers alone are not sufficient; you MUST indicate how you derived them (show your work). When submitting a numerical answer which is a decimal, use the number of decimal places warranted by the data. Some problems are worth more points than others; the value of a problem is indicated in parentheses following the problem number. The exam totals 200 points. Problem 1 (20 pts) . Matilda throws a big party to celebrate the A she got in Math 218. Among many other things she prepared a very special fruit punch with fresh fruits from her garden. The punch is dispensed by a dispensing machine that she received as a gift when she successfully passed Math 118. The amount of fruit punch dispensed varies from cup to cup, and it may be looked upon as a random variable having a normal distribution with a mean of 7 . 7 oz and a standard deviation of 0 . 25 oz. (a) Find the probability that the machine will dispense between 7 . 33 oz and 7 . 51 oz of fruit punch. (b) Find the probability that the machine will dispense more than 8 oz. (c) If 20 cups are filled with fruit punch, what is the expected number of cups that are filled with more than 8 oz? (d) The machine has a knob which allows the mean to be reset, without changing the standard deviation. Matilda wants to reset the machine to make sure that 98 . 5% of the time it will not dispense more than 8 oz. What should the new mean be set to? Problem 2 (20 pts) . The graph of a probability density function f ( x ) is shown below. The function has the properties that it is zero outside the interval [0 , 1]; that f (1 / 2) = 0 and f (0) = f (1) = c , a constant to be determined; that it is linear on the interval [0 , 1 / 2]; that it is linear on [1 / 2 , 1] with equation y = 4 x - 2; and that it is continuous on [0 , 1].
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