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Homework 1 (MATH 4255-01) Due date: Tuesday, Feb. 2, 2010 Name (Print): 1. Temperature measurements (in C) result in the following numbers Ni of temperature values which where found in a certain temperature interval Ti: Ti 8-10 10-12 12-14 14-16 16-18 18-20 20-22 22-24 24-26 Ni 500 1400 2200 3600 5000 3300 2600 1300 300 a) Calculate the mean of the temperature distribution. b) Calculate the standard deviation of the temperature distribution. c) Assume that the temperature distribution can be described by normal distribution. Make a plot of the normal probability density function (PDF) that corresponds to the given data set. d) Calculate approximately the probability P(16 T 22) to find T between 16C and 22C. Solutions: a) <T> = 16.94; b) = 3.5; d) P(16 T 22) = 53.2% 2. According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. a) What is the probability that an adult male chosen at random is between 65 inches and 73 inches tall? b) What percentage of the adult male population is more than 6 feet tall? Solutions: a) P(65 X 73) = 84.7%; b) P(72 X ) = 14.2%; 3. Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. a) If the target weight is 500 g, what is the probability that the machine produces a box with less than 480 g of cereal? b) Suppose a law states that no more than 5% of a manufacturer’s cereal boxes can contain less than the stated weight of 500 g. At what target weight should the manufacturer set its filling machine? Solutions: a) P(0 X 480) = 4.78%; b) <X> = 520 g;