Barnett/Ziegler/Byleen Finite Mathematics 11e1 Learning Objectives for Section 11.5 Normal Distributions The student will be able to identify what is meant by a normal distribution. The student will be able to find the area under normal curves. The student will be able to approximate the binomial distribution with a normal distribution.

Barnett/Ziegler/Byleen Finite Mathematics 11e2 Normal Distributions We have seen that the histogram for a binomial distribution with n = 20 trials and p = 0.50 was shaped like a bell if we join the tops of the rectangles with a smooth curve. Real world data, such as IQ scores, weights of individuals, heights, test scores have histograms that have a symmetric bell shape. We call such distributions normal distributions. They will be the focus of this section.

Barnett/Ziegler/Byleen Finite Mathematics 11e3 Mathematicians and the Normal Curve Three mathematicians contributed to the mathematical foundation for this curve. They are Abraham De Moivre, Pierre Laplace and Carl Friedrich Gauss.

Barnett/Ziegler/Byleen Finite Mathematics 11e4 Abraham De Moivre De Moivre pioneered the development of analytic geometry and the theory of probability. He published The Doctrine of Chance in The definition of statistical independence appears in this book together with many problems with dice and other games. He also investigated mortality statistics and the foundation of the theory of annuitiestheory of probability De Moivre

Barnett/Ziegler/Byleen Finite Mathematics 11e5 Pierre Laplace Laplace systematized and elaborated probability theory in "Essai Philosophique sur les Probabilités" (Philosophical Essay on Probability, 1814). He was the first to publish the value of the Gaussian integral. probabilityGaussian integral We will talk about Gauss later. Laplace

Barnett/Ziegler/Byleen Finite Mathematics 11e6 Bell-Shaped Curves Many frequency distributions have a symmetric, bell shaped histogram. Example 1: The frequency distribution of heights of males is symmetric about a mean of 69.5 inches. Example 2: IQ scores are symmetrically distributed about a mean of 100, with a standard deviation of 15 or 16. The frequency distribution of IQ scores is bell shaped. Example 3: SAT test scores have a bell shaped, symmetric distribution.

Barnett/Ziegler/Byleen Finite Mathematics 11e7 Properties of Normal Curves Normal curves are bell-shaped and are symmetrical with respect to the vertical line x =  (the mean). The curve approaches, but does not touch, the horizontal axis as x gets very large (or x gets very small) The shape of a normal curve is completely determined by its mean and standard deviation - a small standard deviation indicates a tight clustering about the mean and thus a tall, narrow curve; a large standard deviation indicates a large deviation from the mean and thus a broad, flat curve.

Barnett/Ziegler/Byleen Finite Mathematics 11e8 Graphs of Normal Curves Several normal curves

Barnett/Ziegler/Byleen Finite Mathematics 11e9 Probability and Area under the Normal Curve Key fact: For a normally distributed variable, the percentage of observations that lie within a specified range equals the corresponding area under its associated normal curve. This is approximately true for a variable that is approximately normally distributed. p(a < x < b) = probability that the random variable X is between a and b = area under the normal curve between x = a and x = b. The total area under a normal curve is 1.

Barnett/Ziegler/Byleen Finite Mathematics 11e10 Finding Areas Under a Normal Curve Finding the area under a normal curve between x = a and x = b requires calculus. We can circumvent this problem by looking up the values in a table. However, the shape of each normal curve is determined by the standard deviation; the greater the standard deviation, the “flatter” and more spread out the normal curve will be. We would need infinitely many tables. The solution is to standardize a normally distributed variable, and to use the table for the standard normal curve.

Barnett/Ziegler/Byleen Finite Mathematics 11e11 Standard Normal Distribution The standard normal distribution has a mean of 0 and a standard deviation of 1. Values on the horizontal axis are called z values. Values on the y axis are probabilities and are decimal numbers between 0 and 1, inclusive.

Barnett/Ziegler/Byleen Finite Mathematics 11e12 Areas under the Standard Normal Curve For the following examples, 1. Draw a diagram 2. Shade the appropriate area 3. Use a table or a TI 83 to find the probability (A) Find p(0 < z < 1.2)

Barnett/Ziegler/Byleen Finite Mathematics 11e13 Areas under the Standard Normal Curve For the following examples, 1. Draw a diagram 2. Shade the appropriate area 3. Use a table or a TI 83 to find the probability (A) Find p(0 < z < 1.2) Look up the z value of 1.2 Answer:

Barnett/Ziegler/Byleen Finite Mathematics 11e14 Areas (continued) (B) Find p(-1.3 < z < 0)

Barnett/Ziegler/Byleen Finite Mathematics 11e15 Areas (continued) (B) Find p(-1.3 < z < 0) We can ignore the sign of z since the graph is symmetrical. Look up the z value of 1.3. Answer:

Barnett/Ziegler/Byleen Finite Mathematics 11e16 Areas (continued) (C) Find p(-1.25 < z < 0.89)

Barnett/Ziegler/Byleen Finite Mathematics 11e17 Areas (continued) (C) Find p(-1.25 < z < 0.89) Use table to find two different areas, and add: area left of y axis = , area right of y axis = Answer:.7077

Barnett/Ziegler/Byleen Finite Mathematics 11e18 Areas (continued) (D) Find p(z >.75)

Barnett/Ziegler/Byleen Finite Mathematics 11e19 Areas (continued) (D) Find p(z >.75) Use table to find p(0 < z < 0.75) = Subtract this area from Answer:

Barnett/Ziegler/Byleen Finite Mathematics 11e20 Standardizing a Normally Distributed Variable To find p(a < x < b) for a normal curve with mean  and standard deviation , we calculate where The variable z is called the standard normal variable.

Barnett/Ziegler/Byleen Finite Mathematics 11e21 Example Assume IQ scores are distributed normally with a mean of 100 and standard deviation of 16. (A) If the IQ of an individual is x = 124, what z value corresponds to this?

Barnett/Ziegler/Byleen Finite Mathematics 11e22 Example Assume IQ scores are distributed normally with a mean of 100 and standard deviation of 16. (A) If the IQ of an individual is x = 124, what z value corresponds to this? z =

Barnett/Ziegler/Byleen Finite Mathematics 11e23 Example (continued) (B) Find the probability that a randomly chosen person has an IQ greater than 120.

Barnett/Ziegler/Byleen Finite Mathematics 11e24 Example (continued) (B) Find the probability that a randomly chosen person has an IQ greater than 120. Step 1. Draw a normal curve and shade appropriate area. State the probability: p(x > 120) where x is IQ.

Barnett/Ziegler/Byleen Finite Mathematics 11e25 Example (continued) Step 2. Convert x score to a standardized z score: z = (120 – 100) / 16 = 20/16 = 5/4 = 1.25 p(x > 120) = p(z > 1.25) Step 3. Use table or TI 83 to find area. Answer: =

Barnett/Ziegler/Byleen Finite Mathematics 11e26 Example 2 A traffic study at one point on an interstate highway shows that vehicle speeds are normally distributed with a mean of 61.3 mph and a standard deviation of 3.3 miles per hour. If a vehicle is randomly checked, find the probability that its speed is between 55 and 60 miles per hour.

Barnett/Ziegler/Byleen Finite Mathematics 11e27 Step 2. Convert x score to a standardized z score: Step 3. Use table or TI 83 to find area. Answer: = Example 2 (continued) Step 1. Draw a normal curve and shade appropriate area. State the probability: p(55 < x < 60) where x is speed. = p(-1.91 < z < -0.39)

Barnett/Ziegler/Byleen Finite Mathematics 11e28 Mathematical Equation for Bell-Shaped Curves Carl Friedrich Gauss, a mathematician, was probably the first to realize that certain data had bell-shaped distributions. He determined that the following equation could be used to describe these distributions: where ,  are the mean and standard deviation of the data.

Barnett/Ziegler/Byleen Finite Mathematics 11e29 Using the Normal Curve to Approximate Binomial Probabilities Example: Find the probability of getting 12 or more Heads when you toss a coin 20 times.

Barnett/Ziegler/Byleen Finite Mathematics 11e30 Using the Normal Curve to Approximate Binomial Probabilities Example: Find the probability of getting 12 or more Heads when you toss a coin 20 times. Solution: We have seen that the histogram for a binomial distribution with n = 20 trials and p = 0.50 was shaped like a bell if we join the tops of the rectangles with a smooth curve. To find the probability that x (number of heads) is greater than or equal to 12, we would have to calculate p(x=12) + p(x=13) + p(x=14) + … p(x=20). The calculations would be very tedious, to say the least.

Barnett/Ziegler/Byleen Finite Mathematics 11e31 Using the Normal Curve to Approximate Binomial Probabilities We could, instead, treat the binomial distribution as a normal curve, since its shape is pretty close to being a bell-shaped curve, and then find the probability that x is greater than 12 using the procedure for finding areas under a normal curve.  = np = 10  = sqrt(np(1-p)) = sqrt(5) = 2.24 Probability that x ≥ 12 = total area in yellow

Barnett/Ziegler/Byleen Finite Mathematics 11e32 Approximating Binomial by Normal (continued) Because the normal curve is continuous and the binomial distribution is discrete, we have to make what is called a correction for continuity. Since we want p(x ≥ 12), we must include the rectangular area corresponding to x = 12. The base of this rectangle starts at 11.5 and ends at Therefore, we must find p(x > 11.5). The rectangle representing the p(x = 12) extends from 11.5 to 12.5 on the horizontal axis.

Barnett/Ziegler/Byleen Finite Mathematics 11e33 Approximating Binomial by Normal (continued) Using the procedure for finding area under a non-standard normal curve we get = p(z > 0.770) = =

Barnett/Ziegler/Byleen Finite Mathematics 11e34 Rule of Thumb Test Use a normal distribution to approximate a binomial distribution only if the interval lies entirely in the interval from 0 to n. (n is sample size.)