1/14 MTH 245: Mathematics for Management, Life, and Social Sciences May 18, 2015 Section 7.6 Section 7.6: The Normal Distribution. 2/14 The Normal Distribution. Figure: Abraham DeMoivre
Section 7.6: The Normal Distribution. 3/14 The normal curve Abraham DeMoivre proved that areas under the curve f z (z) = 1 e z2 /2, 2π can be used to estimate P a X n( ) 1 2 n ( )( 12 ) b, where X is a random 12 binomial random variable with a large n and p = 1 2. The curve is called normal curve. We will take a look into continuous probability by studying experiments with normally distributed outcomes. For instance, we might consider choosing a newborn and observing his or her weight, chose a college student on campus and observe his or her height, etc. Associated to each experiment is the normal curve, a bell-shaped curve. Section 7.6: The Normal Distribution. 4/14 0.2 0.15 0.1 0.05 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Section 7.6: The Normal Distribution. 5/14 Probability for a normal distribution The probability that the value of the random variable, X, lies between two values, a and b, is the fraction of the area under the normal curve that lies between x = a and x = b, we denote it by P(a X b). The total area under the normal curve is always 1. A normal curve is completely described by its mean µ and standard deviation. Given µ and we write down the equation of the associated normal curve as y = 1 2π e ( 2)( 1 x µ ) 2. The standard normal curve has mean µ = 0 and standard deviation (s.d.) = 1. Section 7.6: The Normal Distribution. 6/14 0.8 0.6 0.4 0.2 0 2 1 0 1 2 3
Section 7.6: The Normal Distribution. 7/14 Let Z be a random variable having the standard normal distribution. Let z be any number, A(z) denotes the area under the standard normal curve to the left of z If X is a random variable having a normal distribution with mean µ and standard deviation, then ( a µ P(a X b) = P Z b µ ) ( ) ( ) b µ a µ = A A Section 7.6: The Normal Distribution. 8/14 and ( P(X x) = P Z x µ ) ( ) x µ = A, where Z has the standard normal distribution and A(z) is the area under that distribution to the left of z. z-scores allow us to use a table to find the amount of area that is to the left of a given value,x, in a normal distribution. value of x mean Z score = standard deviation = x µ. The z-score should be interpreted as the number of standard deviations above/below the mean.
Section 7.6: The Normal Distribution. 9/14 Appendix A Section 7.6: The Normal Distribution. 10/14 Figure: Areas under the Standard Normal Curve Example 1 For the case where x = 38.4, µ = 22.5, and = 6.2 a) Find the z-score. b) What percent of the area of the normal curve lies below x = 38.4?
Section 7.6: The Normal Distribution. 11/14 Example 2 Find the area under the normal curve with µ = 7, = 2 from x = 6 to x = 10. This represents P(6 X 10) for a random variable X having the given normal distribution. Section 7.6: The Normal Distribution. 12/14 Definition 1 If a score S is in the pth percentile of a normal distribution, then p% of all scores fall bellow S, and (100 p)% of all scores fall above S. Example 3 What is the 90th percentile of the standard normal distribution?
Section 7.6: The Normal Distribution. 13/14 SAT scores Example 4 Assume that SAT verbal scores for a first-year class at a university are normally distributed with mean 520 and standard deviation 75. (a) The top 10% of the students are placed into the honors program for English. What is the lowest score for admittance into the honors program? (b) What is the range of the middle 90% of the SAT verbal scores at the university? (c) Find the 98th percentile of the SAT verbal scores. Section 7.6: The Normal Distribution. 14/14 Example 5 The lifetime of a certain brand of tires ins normally distributed with mean µ = 30,000 miles and standard deviation = 5000. The company has decided to issue a warranty for the tires but does not want to replace more than 2% of the tires that it sells. At what mileage should the warranty expires?