15.5 The Normal Distribution

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 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 *Good strategy is to sketch the normal curve & shade the area you are looking for 68% 95% 99.7%

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 = 100  = 15 µ –  = 100 – 15 = 85 µ +  = = 115 µ  85115µ Both scores are within 1 SD of mean  68% b) Above 115? 115 µ µ +  = = – 84 = 16%

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  = (15) = % % 100 – 2.5 = 97.5% 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 = 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 = –.5 =.3485 b) between z = – 0.9 and z = 0.9 want: table: z = –.5 = =.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 = –.7910 =.2090 d) to the right of z = – 0.81 want: same area! want.7910 table: z =.81

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?.6700 table: z =.44 want: use symmetry 1 –.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 #1505 Pg 829 #1 – 11, 17 – 23, 25 – 28