Data Analysis & Probability

Transcription

1 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= and Z = 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*

2 Example 2: Sketch and find the area under the standard normal curve to the RIGHT of Z = 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 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

3 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.

4 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= 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?

5 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= Determine the area under the standard normal curve that lies to the left of Z = Determine the area under the standard normal curve that lies to the right of Z= Determine the area under the standard normal curve that lies between Z = and Z = Determine the area under the standard normal curve that lies to the right of Z = Determine the area under the standard normal curve that lies between Z = and Z = Determine the area under the standard normal curve that lies to the left of Z = -2 or to the right of Z = 2.

6 8. Determine the area under the standard normal curve that lies to the left of Z = or to the right of Z = Find the Z-score such that the area under the standard normal curve to the left of Find the Z-score such that the area under the standard normal curve to the right is Find the Z-score such that the area under the standard normal curve to the right is Find the Z-score such that the area under the standard normal curve to the left is 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.