How do I use normal distributions in finding probabilities? Thursday, November 4 Essential Questions How do I use normal distributions in finding probabilities?

Use Normal Distributions 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% 34% x – s + s – 2s + 2s – 3s + 3s 68% 95% 13.5% 13.5% 2.35% 2.35% 99.7% x – 3s x – 2s x – s x + s x + 2s x + 3s 0.15% 0.15% x

Use Normal Distributions 7.4 Use Normal Distributions Example 1 Find a normal probability x – s + s – 2s + 2s – 3s + 3s A normal distribution has a mean x and standard deviation s. For a randomly selected x-value from the distribution, find Solution The probability that a randomly selected x-value lies between _______ and _________ is the shaded area under the normal curve. Therefore:

Use Normal Distributions 7.4 Use Normal Distributions Checkpoint. Complete the following exercise. A normal distribution has mean x and standard deviation s. For a randomly selected x-value from the distribution, find x – s + s – 2s + 2s – 3s + 3s

Use Normal Distributions 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 scores between 387 and 605? 169 278 387 496 605 714 823 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

Use Normal Distributions 7.4 Use Normal Distributions Checkpoint. Complete the following exercise. 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

Use Normal Distributions 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.

Use Normal Distributions 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 The table shows that P(z < ____) = _______. z .0 .1 .2 -3 .0013 .0010 .0007 -2 .0228 .0179 .0139 -1 .1587 .1357 .1151 -0 .5000 .4602 .4207 .5398 .5793 1 .8413 .8643 .8849 So, the probability that a randomly selected test-taker received a math score of at most 630 is about ________.

Use Normal Distributions 7.4 Use Normal Distributions Checkpoint. Complete the following exercise. In Example 3, find the probability that a randomly selected test-taker received a math score of at most 620?

Use Normal Distributions 7.4 Use Normal Distributions Pg. 277, 7.4 #1-21