Lesson 15-5 The Normal Distribution Today’s Date: 5/4/17 Lesson 15-5 The Normal Distribution Pg 829 #1 – 11, 17 – 23, 25 – 28 Pre-calculus Normal Distribution (Bell Curve) Z-Table

A frequency polygon can be replaced by a smooth curve A data set that is normally distributed is called a normal curve. The normal curve: - has the general shape of a bell (aka bell curve) - is symmetric about the vertical line through the mean - approaches the horizontal axis at both extremes - mean = median = mode at the line of symmetry Standard deviation measures how the data varies from the middle. The total area under a normal curve is 1 - because the curve represents the total probability distribution. * Note µ (mu) will represent the mean of the data set.

For any normal distribution, the following holds (empirical rule): About 68% of the data lies within 1 standard deviation of the mean About 95% of the data lies within 2 standard deviations of the mean About 99.7% of the data lies within 3 standard deviations of the mean 68% 95% 99.7% *Good strategy is to sketch the normal curve & shade the area you are looking for

Ex 1) For the general population, IQ scores are normally distributed with a mean of 100 and standard deviation of 15. Approximately what percent of the population have IQ scores: a) Between 85 and 115? µ  µ = 100  = 15 µ –  = 100 – 15 = 85 µ +  = 100 + 15 = 115 Both scores are within 1 SD of mean  68% 85 µ 115 b) Above 115? µ +  = 100 + 15 = 115 50 34 100 – 84 = 16% µ 115

Ex 1) For the general population, IQ scores are normally distributed with a mean of 100 and standard deviation of 15. Approximately what percent of the population have IQ scores: c) Below 130? µ + 2 = 100 + 2(15) = 130 100 – 2.5 = 97.5% 130 2.5% 95% 2.5%

Ex 2) In her first year at college, Sandy received a grade of 80 in math and 74 in psychology. The 80 in math was in a class with a mean of 72 and a standard deviation of 4. The 74 in psychology was in a class with a mean of 66 and a standard deviation of 8. Which grade is relatively better? *Determine how many standard deviations above the mean each score is math  psychology  Math grade is relatively better

What if the value we wanted to investigate was not exactly at a standard deviation? We can use a table of values using the z-score. Normal Distribution Tables Table shows value of area under the normal curve to the left of z For example, if z = 0.31 then area under the curve = 0.6217 What if your z-score value is negative? Use symmetry and draw that picture!

Ex 3) Using the table, find the indicated areas under the standard normal curve. a) between z = 0 and z = 1.03 want: table: z = 1.03 .8485 – .5 = .3485 .8485 b) between z = – 0.9 and z = 0.9 want: table: z = 0.9 .8159 – .5 = .3159 .8159 .3159 + .3159 = .6318

Ex 3) Using the table, find the indicated areas under the standard normal curve. c) to the right of z = 0.81 want: table: z = .81 1 – .7910 = .2090 .7910 want d) to the right of z = – 0.81 want: table: z = .81 .7910 same area!

Ex 4) An automobile manufacturer claims that a new car gets an average 42 mpg and that the mileage is normally distributed with a standard deviation of 4.5 mpg. If this claim is true, what percent of these new cars will get less than 40 mpg? table: z = .44 want: use symmetry 1 – .6700 = .3300 33% .6700 ***NOTE: This question could have been phrased, “What is the probability that one of these cars will have mileage less than 40 mpg?” (answer would be .3300)

Homework Pg 829 #1 – 11, 17 – 23, 25 – 28