Unit 5 Probability Distributions Name: Date: Hour: Section 7.2: The Standard Normal Distribution (Area under the curve) Notes By the end of this lesson, you will be able to Find the area under the standard normal curve. Find z-scores for a given area Interpret the area under the standard normal curve as a probability Review: ~ If X is a normally distributed random variable, we can use the under the normal density function to find the that a randomly selected individual from the population has a certain characteristic. ~ We can relate a normal random variable to the standard normal random variable through the Z-score Z The STANDARD normal curve is the one with mean μ = 0 and standard deviation σ = 1. ~ It has the same properties as any normal curve - inflection points are at µ + σ and µ - σ - the graph is symmetric about the mean - Its highest point is the mean - the area under the curve is 1 - The Empirical Rule applies Finding the Area Under the Curve when given Z-Scores. Example 1: Find the area under the standard normal curve between Z= -0.23 and Z = 1.68. Step 1: First, draw a standard normal curve. Step 2: USE YOUR CALCULATOR! Press 2 nd VARS Select 2: normalcdf( Then enter your boundsseparated by commas: normaldcf(lower bound, upper bound) Hit Enter NOTE: No lower bound? Use -1E99 No upper bound? Use 1E99 *Pressing 2 nd, (comma) gets you the E*
Example 2: Sketch and find the area under the standard normal curve to the RIGHT of Z = -0.46. Example 3: Sketch and find the area under the standard normal curve to the LEFT of Z=2.23. Finding a Z-score from a Specified Area Example 4 (Specified area to the LEFT): Find the Z-score so that the area to the left of the Z-score is.4332. Step 1: USE YOUR CALCULATOR! Press 2 nd VARS Select 3: invnorm( Then enter your area to the left: invnorm( area to the left ) Hit Enter Example 5 (Specified area to the RIGHT): Find the Z-score so that the area to the right of the Z-score is.0351
Example 6: Find the Z-score that separates the lower 60% from the upper 40%. Example 7: Find the Z-score for which 43.1% of the data is to the right of. Example 8: Find the value of z0.10 **This notation means: Find the z-score so that the area under the standard normal curve to the right is 0.10.
Quick Check Section 7.2: The Standard Normal Distribution (Area under the curve) Self-Assessment 1. Sketch and find the area under the curve to the right of the Z=0.4332. 2. Sketch and find the value of z0.21. Learning Goals Find the area under the standard normal curve. Find z-scores for a given area Self-Assessment I am unsure of or confused about this I am ready to start practicing I am already good at this Interpret the area under the standard normal curve as a probability My Goals for Today- thinking about what I am good at, where am I confused and what do I need to work on? What do I do if I am confused or need to work on a learning target?
Name: Date: Hour: Unit 5 Probability Distributions Section 7.2: The Standard Normal Distribution (Area under the curve) Homework ROUND ALL AREAS TO FOUR DECIMAL PLACES AND Z-SCORES TO 2 DECIMAL PLACES 1. Determine the area under the standard normal curve that lies to the left of Z= -2.45. 2. Determine the area under the standard normal curve that lies to the left of Z = 3.49. 3. Determine the area under the standard normal curve that lies to the right of Z=-3.01. 4. Determine the area under the standard normal curve that lies between Z = -0.55 and Z = 0. 5. Determine the area under the standard normal curve that lies to the right of Z = 3.11 6. Determine the area under the standard normal curve that lies between Z = -1.67 and Z = 1.98. 7. Determine the area under the standard normal curve that lies to the left of Z = -2 or to the right of Z = 2.
8. Determine the area under the standard normal curve that lies to the left of Z = -0.88 or to the right of Z = 1.23. 9. Find the Z-score such that the area under the standard normal curve to the left of 0.98. 10. Find the Z-score such that the area under the standard normal curve to the right is 0.25. 11. Find the Z-score such that the area under the standard normal curve to the right is 0.75. 12. Find the Z-score such that the area under the standard normal curve to the left is 0.1. 13. Find the Z-score that separates the lower 20% and upper 80% of data. 14. Find the Z-score that 33.3% of the data lies to the left of. 15. Find the Z-score that 67% of the data lies to the right of.