How do I use normal distributions in finding

• 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 ____. • About ___% of the area lies within 1 standard deviation of the mean. • About ___% of the area lies within 2 standard deviation of the mean. • About _____% of the area lies within 3 standard deviation of the mean. 34% 68% 95% 99. 7% 34% 13. 5% 2. 35% 0. 15% – 2 s –s + 3 s – 3 s + 2 s + 3 s +s + 2 s –s +s – 2 s x x x x – 3 s 0. 15%

7. 4 Use Normal Distributions +s + 2 s + 3 s –s x – 2 s x x x A normal distribution has a mean x and standard deviation s. For a randomly selected x-value from the distribution, find – 3 s Example 1 Find a normal probability Solution The probability that a randomly selected x-value lies between _______ and _____ is the shaded area under the normal curve. Therefore:

7. 4 Use Normal Distributions Checkpoint. Complete the following exercise. + 2 s + 3 s –s +s – 2 s x x x x – 3 s 1. A normal distribution has mean x and standard deviation s. For a randomly selected x-value from the distribution, find

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 ____ standard deviation on either side of the mean. So the percent of test-takers with scores between 387 and 605 is

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

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.

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 z . 0 . 1 . 2 -3 . 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 < ____) = _______. So, the probability that a randomly selected test-taker received a math score of at most 630 is about ____.

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?

7. 4 Use Normal Distributions Pg. 277, 7. 4 #1 -21