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GPS PreCalculus Day 75 Notes – Normal Distributions Normal Distribution: a probability distribution with the mean( X ) and standard devitation ( s X ) modeled by a bell shaped curve. The term Normal Curve refers to a smooth, symmetrical bell-shaped curve. 34% 34% 13.5% 0.15% 13.5% 2.35% X - 3s X 2.35% X - 2s X X - 1s X X + 1s X X X + 2s X 0.15% X + 3s X Important info about normal distributions: 1. 50% of the data lies above the mean, 50% of the data lies below the mean 2. 68% of the data lies between -1 and +1 standard deviations from the mean 3. 95% of the data lies between -2 and +2 standard deviations from the mean 4. 99.7% of the data lies between -3 and +3 standard deviations from the mean Standard Normal Distribution: a normal distribution with a mean = 0 and a standard deviation = 1 34% 34% 13.5% 2.35% 0.15% Z-scores 13.5% -3 -2 2.35% -1 0 +1 +2 0.15% +3 Z-score: the number z of standard deviations that a data value lies above or below the mean z= x- X sX –1 GPS PreCalculus Day 75 Notes – Normal Distributions Example 1: A set of data is normally distributed with a mean( X ) and standard devitation ( s X ) . a. P(X - s X Ј x Ј X + 2s X ) This question is asking what percentage of the data lies between -1 standard deviation and +2 standard deviations. Since it is a NORMAL DISTRIBUTION, we can use the percentages from the normal distribution curve So, we add up the percentages from -1 standard deviation to +2 standard deviations: 34% + 34% + 13.5% = 81.5% Final Answer: 81.5% b. P( x Ј X + 2s X ) This question wants to know what percentage is less than +2 standard deviations from the mean. Again, using our normal distribution curve, we add up all the percentages LESS THAN +2 standard deviations: 13.5% + 34% + 34% + 13.5% + 2.35% + 0.15% = 97.5% Final Answer: 97.5% Example 2: Math scores of an exam are normally distributed with a mean = 496 and a standard deviation = 109 a. b. c. What percentage scored between 387 and 605? What percentage scored less than 278? Find the z-score for a test score of 630? Solution: First, we must draw a normal distribution and label our x-axis appropriately 34% 34% 13.5% 0.15% 169 13.5% 2.35% 2.35% 278 387 Mean-1SD: 496-109 496 Mean 605 714 0.15% 823 Mean+1SD: 496+109 Solution (a): Add up the percentages from 387-605: 34% + 34% = 68% Solution (b): Add up the percentages less than 278: 2.35% + 0.15% = 2.5% Solution (c): Use the formula to find a z-score: z = x - X 630 - 496 = = 1.2 sX 109 Assignment: Complete WS 8.2 and TURN-IN –2