Introduction to the Normal Distribution (Dr. Monticino)

Assignment Sheet Math 1680  Read Chapter 5  Assignment #5  Due Monday, Feb. 21 st  Chapter 5  Set A: 1,2  Set B: 1,2,3,5  Set C: 1,2,3  Set D: 1,2,3,4,5  Set E: 1,2,3  Review exercises 1,2,3,4,7,8

Overview  Motivation  Calculations

The Normal Distribution  Histograms for a variety of data often have a “bell curve” shape  Because of this empirical observation and theoretical results, often approximate the distribution of a population by a normal distribution  Recall area of bars in histogram give percentage in a group.  Similarly, the area under a specified portion of the normal distribution curve gives the percentage in that range

Normal Distribution Approximation

The Normal Distribution  The standard normal curve is symmetric about 0  50% of the area is to the left of 0 and 50% to the right of 0  Analogous areas on opposite sides of 0 have the same area 50% same area

Standard Normal Distribution  68% of the area is within [-1,1]; 95% between [-2,2]; and 99.7% between [- 3,3]  Other values are given by a “normal table”

Converting to SD Units  Examples  Suppose that the average weight for U.S. men is 175 lbs. and the standard deviation for weight is 20 lbs.  If a man weighs 190 lbs., how many standard deviation units away from the mean weight is he?  A man is 3.2 standard deviations above the mean- weight. How heavy is he?  Ninety-five percent of U.S. men weigh between what two values?

Examples  Find area under normal curve  to the right of 1  to the left of -.5  to the left of.75  between -1.5 and 1.5  outside of -2 and 2

Standard Normal Table

Examples  The birthweight for babies is approximately normally distributed with an average birthweight of 7.2 lbs. and a standard deviation of 1.1 lbs.  A newborn baby weighing 9 lbs is how many standard deviation units away from the mean weight?  What percent of newborns weigh between 8.3 and 6.1 lbs.  What percent of babies weigh more than 9 lbs?  At what percentile of weight is a 9 lb. baby?  What is the 25 th percentile for newborns?  Thirty percent of babies weigh between what two values (centered on the mean)?

More Examples  Nationally, SAT scores are distributed normally with an average score of 1000 and a standard deviation of 150  What percent of students taking the SAT score above 1400  What percent score between 900 and 1200  What is the 95 th percentile of SAT scores (Dr. Monticino)