Normal Distributions MM2D1d Compare the means and standard deviations of random samples with the corresponding population parameters, including those population parameters for normal distributions. Observe that the different sample means vary from one sample to the next. Observe that the distribution of the sample means has less variability than the population distribution.

Normal Distributions Normal Distribution – modeled by a bell- shaped curve called a normal curve. The total area under the curve is 1.

Normal Distributions Another breakdown

Normal Distributions Empirical Rule-only applies for normal distributions and applies to the normal bell curve. ____ of the data will be located within one standard deviation symmetric to the mean. ____ of the data will be located within two standard deviations …. ____of the data will be located within three standard deviations…. 68% 95% 99.7%

Normal Distributions Find a normal probability For these samples REMEMBER 1 standard deviation is 68% which means 34% for -1 and 34% for standard deviations is 95% which means 47.5% for -2 and 47.5% for standard deviations is 99.7% which means 49.85% for -3 and 49.85% for +3.

Normal Distributions Construct a bell curve 1.Find the mean of the data. 2.Find the standard deviation of the data. 3.The mean goes in the middle of the curve. 4.The standard deviation is added or subtracted for each interval.

Normal Distribution Bell curves Test scores 86, 88, 90, 99, 70 Mean 86.6 Standard deviation 9.41 The bottom of the curve will be labeled x-3σ x-2σ x-1σ x x+1σ x+2σ x+3σ

Normal Distributions Normal Probability Mean +2 standard deviations mean how much percent? What is that in decimal form? 47.5%.475

Normal Distributions Normal Probability Try one: 84%

Normal Distributions Interpret Normally Distributed Data The heights (in feet) of fully grown white oak trees are normally distributed with a mean of 90 ft and a standard deviation of 3.5 feet. About what percent of white oak trees have heights less than 97 feet? How about between 83 feet and 90 feet? 97.5% 47.5%

Normal Distributions Using z-scores and the standard normal table Standard normal distribution-mean 0 and standard deviation 1. Formula that can turn x values from normal distributions into z values Z values for a particular x value is called the z- score (the number of standard deviations the x- value lies above or below the mean.)

Normal Distributions Using z-scores and the standard normal table Find the probability that a randomly selected white oak tree has a height of at most 94 feet. (remember 90 is the mean from a previous problem. Find the z-score corresponding to the x- value of 94 feet. (Page 265) Use the formula 94-90/3.5=1.1

Normal Distributions Using z-scores and the standard normal table Try one: Find the probability that a randomly selected white oak tree has a height of at most 85 feet. Use the table on page 296 in your book. About

Normal Distributions Page 267 (1-20)

Normal Distributions Exploration Materials: Laptops Instructions: Go to at least three of the websites listed. Take notes on the presentations. Make sure your notes include at least three examples of problems worked out. Write a review of the websites by ranking them from 1 to 3 with 3 being the most helpful. Tell why you ranked them as you did. Find two more web based resources on the lesson and include them on your sheet to turn in.

Normal Distribution Web search 8.htmlhttp://mathforum.org/library/drmath/view/ html mal/Normal%20DistributionShow.ppshttp:// mal/Normal%20DistributionShow.pps ppthttp://cda.morris.umn.edu/~benw/ppt/1.3.normal. ppt alDistribution4.PPThttp://gunston.gmu.edu/healthscience/597/Norm alDistribution4.PPT 411_sp07/ch9.ppthttp:// 411_sp07/ch9.ppt