Normal Distributions and the Empirical Rule Learning Goal: Students will be able to recognize normal distributions and use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. CRM 3.4 Honors - Lesson 04

Normal Distributions and the Empirical Rule A bell-shaped, symmetric distribution with a tail on each end is called a normal distribution. notes modified from HMH - Algebra 1 ACE © 2015 Lesson 13.5 p. 464 notes and examples modified from Math Nation, Section 9 - Topic 8

Normal Distributions and the Empirical Rule We can use the empirical rule to understand the data distribution. The empirical rule states that normal distributions approximately fall within the following percentages: notes modified from HMH - Algebra 1 ACE © 2015 Lesson 13.5 p. 464 notes and examples modified from Math Nation, Section 9 - Topic 8

Normal Distributions and the Empirical Rule Example: Assume that we have a data set so large that we are not given a list of all the values. We are told the data follows a normal distribution with a mean of 16 and standard deviation of 4. Label the distribution below with the mean and three standard deviations above and below the mean. Suppose one of the data values is 20. An observation of 20 is ___ standard deviation(s) _______ the mean. Suppose one of the data values is 8. An observation of 8 is ___ standard deviation(s) _______ the mean. For this normal distribution: Approximately 68% of the values will fall between ____ and ____ according to the empirical rule. Approximately 95% of the values will fall between ____ and ____ according to the empirical rule. Approximately 99.7% of the values will fall between ____ and ____ according to the empirical rule.

Normal Distributions and the Empirical Rule: Answer Example: Assume that we have a data set so large that we are not given a list of all the values. We are told the data follows a normal distribution with a mean of 16 and standard deviation of 4. Label the distribution below with the mean and three standard deviations above and below the mean. Suppose one of the data values is 20. An observation of 20 is one standard deviation(s) above the mean. Suppose one of the data values is 8. An observation of 8 is two standard deviation(s) below the mean. For this normal distribution: Approximately 68% of the values will fall between 12 and 20 according to the empirical rule. Approximately 95% of the values will fall between 8 and 24 according to the empirical rule. Approximately 99.7% of the values will fall between 4 and 28 according to the empirical rule.

Normal Distributions and the Empirical Rule Task 1: Suppose the amounts of water a machine dispenses into plastic bottles has a normal distribution with a mean of 16.2 ounces and a standard deviation of 0.1 ounces. Label the mean and three standard deviations above and below the mean on the normal curve below. Then use the normal distribution to complete parts A-C. Part A: The middle 95% of bottles contain between _____ and _____ ounces of water. Part B: Approximately 68% of the bottles contain between _____ and _____ ounces of water. Part C: Approximately 99.7% of the bottles contain between _____ and _____ ounces of water.

Normal Distributions and the Empirical Rule: Answer Task 1: Suppose the amounts of water a machine dispenses into plastic bottles has a normal distribution with a mean of 16.2 ounces and a standard deviation of 0.1 ounces. Label the mean and three standard deviations above and below the mean on the normal curve below. Then use the normal distribution to complete parts A-C. Part A: The middle 95% of bottles contain between 16.0 and 16.4 ounces of water. Part B: Approximately 68% of the bottles contain between 16.1 and 16.3 ounces of water. Part C: Approximately 99.7% of the bottles contain between 15.9 and 16.5 ounces of water.

Using the Empirical Rule to Predict Other Probabilities Let’s use what we’ve learned about the empirical to determine how many data values would fall between each standard deviation in a normal distribution.

Using the Empirical Rule to Predict Other Probabilities Let’s use what we’ve learned about the empirical to determine how many data values would fall between each standard deviation in a normal distribution.

Normal Distributions and the Empirical Rule: Your Turn SAT mathematics scores for a particular year are approximately normally distributed with a mean of 510 and a standard deviation of 80. Label the mean and three standard deviations above and below the mean on the normal curve below. Then use the normal distribution to complete parts A-D. Part A: What percentage of students scored between 350 and 670 points? Part B: What is the probability that a randomly selected score is less than or equal to 510 points? Part C: What is the probability that a randomly selected score is greater than 590 points? Part D: What is the probability that a randomly selected score is greater than 670 points?

Normal Distributions and the Empirical Rule: Answer SAT mathematics scores for a particular year are approximately normally distributed with a mean of 510 and a standard deviation of 80. Label the mean and three standard deviations above and below the mean on the normal curve below. Then use the normal distribution to complete parts A-D. Part A: What percentage of students scored between 350 and 670 points? 95% Part B: What is the probability that a randomly selected score is less than or equal to 510 points? 50% Part C: What is the probability that a randomly selected score is greater than 590 points? 16% Part D: What is the probability that a randomly selected score is greater than 670 points? 2.5%