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# Class Note for MATH 1313 with Professor Heeth at UH

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Date Created: 02/06/15

Math 1313 Section 86 Applications of the Normal Distribution We have reached the last section of the text that we will cover in the course This section covers some word problems involving normally distributed random variables You may notice that there is some similarity between the wording of these problems and the wording of the Cheby chev s Inequality problems You will know to use the normal distribution to solve a problem when you are told that the random variable is normally distributed Ifthose words aren t there then you are probably looking at a Chebychev s Inequality problem Example 1 According to the data released by the Chamber of Commerce the weekly wages of factory workers in a small city are normally distributed with a mean of 1000 and a standard deviation of 100 What is the probability that a randomly selected factory worker who lives in the city makes a weekly wage of a less than 900 b more than 1200 c between 850 and 1150 Example 2 The heights of a certain species of plant are normally distributed with a mean of 30 and a standard deviation of 3 What is the probability that a plant chosen at random will be between 24 and 32 inches tall Sometimes we can use a normal distribution to approximate a binomial distribution We will want to do this when the number of trials of the binomial distribution gets larger Suppose we are given a binomial distribution associated with a binomial experiment involving n independent trials each with a probability of success p and a probability of failure q Then if n is large and p is not close to 0 or 1 the binomial distribution can be approximated by a normal distribution with u np and a 1 n q Using this mean and standard deviation these problems work just like the last few except for ONE LITTLE TWIST When we are approximating the binomial distribution by the normal distribution we are actually estimating the area of the rectangles in the histogram by the area under a normal curve But for X 5 PX 5 is represented by a rectangle that runs from 45 to 55 with height PX 5 So the area for less than five successes actually starts at 45 not 5 Here are two examples In the first n 15 and in the second n 6 Numml PDF r Bhiou al PDF Flyj lh39l I 1 k My particular technique for keeping this straight I Start with the binomial random variable X and write the probability in question in the form PX 2 a or PX lt b or Pa lt X 5 b as determined by the question 2 Then create a normal random variable Y that gives the appropriate edges for the rectangles in the histogram we wish to add the areas of So for example if my question asks for P4 S X lt 9 I want to include the rectangle centered at 4 but not the one centered at 9 ie Iwant to estimate the areas of the rectangles at 4 5 6 7 and 8 My new normal variable becomes P35 lt Y lt 85 3 Then follow the procedure above to turn the normal variable into a standard normal variable so we can read the probability off the chart P35 ltYlt 85P33957ltZlt83957 u 039 039 4 Calculate the numbers above and use the looking up the probability on the standard normal distribution table techniques for the last section Example 3 A company claims that 35 of the households in a certain community use their Squeaky Clean cleanser In a neighborhood of 10 households what is the probability that between 40 and 45 households use the cleanser This is a binomial experiment Use the normal distribution to approximate the probability Example 4 Use the normal distribution to approximate the binomial distribution Suppose that a coin is weighted so that the probability of obtaining a head on a single toss is 03 If the coin is tossed 25 times what is the probability of getting a fewer than 10 heads b Between 9 and 12 heads inclusive 0 More than 8 heads

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