How do I use normal distributions in finding probabilities?

7.4Use Normal Distributions Standard Deviation of a Data Set x x –  x +  x – 2  x + 2  x – 3  x + 3  68% A normal distribution with mean x and standard deviation  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. 95% 99.7% 34% 13.5% 2.35% 0.15% x x –  x +  x – 2  x + 2  x – 3  x + 3  0.15%

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

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

7.4Use 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? 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

2.In Example 2, what percent of the test-takers received scores between 496 and 714? 7.4Use Normal Distributions Checkpoint. Complete the following exercise % 13.5%

7.4Use 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.

7.4Use 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       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 ________.

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

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