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• How do I use normal distributions in finding probabilities? 7.4 Use Normal Distributions Standard Deviation of a Data Set A normal distribution with mean x and standard deviation s has these properties: • The total area under the related normal curve is ____. 1 • About ___% 68 of the area lies within 1 standard deviation of the mean. • About ___% 95 of the area lies within 2 standard deviation of the mean. • About _____% 99.7 of the area lies within 3 standard deviation of the mean. 34% 68% 95% 99.7% 34% 13.5% 2.35% 13.5% 2.35% 0.15% – 2s –s + 3s – 3s + 2s + 3s +s + 2s –s +s – 2s x x x x x x x x x x x x x x – 3s 0.15% 7.4 Use Normal Distributions +s + 2s + 3s x –s x – 2s x x x x Px s x x 2s x A normal distribution has a mean x and standard deviation s. For a randomly selected x-value from the distribution, find – 3s Example 1 Find a normal probability Solution The probability that a randomly selected x-value lies between _______ x 2s is the shaded area x s and _________ under the normal curve. Therefore: Px s x x 2s _____ 0.34 _____ 0.34 ______ 0.135 _______ 0.815 7.4 Use Normal Distributions Checkpoint. Complete the following exercise. 1. A normal distribution has mean x and standard deviation s. For a randomly selected x-value from the distribution, find P x x s . +s + 2s + 3s x –s x x x x x x 0.50 0.34 0.16 – 2s – 3s 7.4 Use Normal Distributions Example 2 Interpret normally distributed data The math scores of an exam for the state of Georgia are normally distributed with a mean of 496 and a standard deviation of 109. About what percent of the test-takers received 169 scores between 387 and 605? 278 387 496 605 714 823 Solution The scores of 387 and 605 represent ____ one standard deviation on either side of the mean. So the percent of test-takers with scores between 387 and 605 is ___ 34 % ____ 34 % ____ 68 % 7.4 Use Normal Distributions Checkpoint. Complete the following exercise. 2. In Example 2, what percent of the test-takers received scores between 496 and 714? 34% 13.5% 169 278 387 496 605 714 823 34 13.5 47.5% 7.4 Use Normal Distributions Example 3 Use a z-score and the standard normal table In Example 2, find the probability that a randomly selected test-taker received a math score of at most 630? Solution Sep 1 Find the z-score corresponding to an x-value of 630. 630 496 z _____ 1.2 s 109 xx __________ __________ _ 7.4 Use Normal Distributions Example 3 Use a z-score and the standard normal table In Example 2, find the probability that a randomly selected test-taker received a math score of at most 630? Solution Sep 2 Use the standard normal table to find Px 630 Pz ___ 1.2 z .0 .1 .2 3 .0013 .0010 .0007 2 .0228 .0179 .0139 1 .1587 .1357 .1151 0 .5000 .4602 .4207 0 .5000 .5398 .5793 1 .8413 .8643 .8849 The table shows that P(z < ____) 1.2 = ._______. 8849 So, the probability that a randomly selected test-taker received a math score of at most 630 is about ________. .8849 7.4 Use Normal Distributions Checkpoint. Complete the following exercise. 3. In Example 3, find the probability that a randomly selected test-taker received a math score of at most 620? 620 496 z s 109 xx Px 620 Pz 1.1 1.1 0.8643 7.4 Use Normal Distributions Pg. 277, 7.4 #1-21