1. Simulations and Normal Distribution Week 4

2. Simulations Probability Exploration Tool

3. Simulations A simulation is made up of  A event called a component that is repeated.  Each component has a finite collection of results that happen at random.  A trial is the occurrence of one component.

4. Creating a Simulation What are you simulating? What is being repeated? How you will assign the random numbers to accurately simulate the distribution of results Record the response variable. What are you counting Run the trials. Analyze (graph) the results State your conclusion.

5. Assigning the Numbers How many digits do you need? What numbers will be a success? What numbers will be a failure? Are there numbers you will ignore? Do the percentages work out?

6. The Normal Model Density Curves and Normal Distributions

7. A Density Curve: Is always on or above the x axis Has an area of exactly 1 between the curve and the x axis Describes the overall pattern of a distribution The area under the curve above any range of values is the proportion of all the observations that fall in that range.

8. Normal Curves Bell shaped, Symmetric,Single-peaked Mean = µ Standard deviation = Notation N(µ, ) One standard deviation on either side of µ is the inflection points of the curve

9. 68-95-99.7 Rule 68% of the data in a normal curve is within one standard deviation of the mean 95% of the data in a normal curve is within two standard deviations of the mean 99.7% of the data in a normal curve is within three standard deviations of the mean

10. Why are Normal Distributions Important? Good descriptions for many distributions of real data Good approximation to the results of many chance outcomes Many statistical inference procedures are based on normal distributions work well for other roughly symmetric distributions

11. Standard Normal Curve

12. Standardizing (z-score) If x is from a normal population with mean equal to µ and standard deviation,  then the standardized value z is the number of standard deviations x is from the mean The unit on z is standard deviations

13. Standard Normal Distribution A normal distribution with µ = 0 and  = 1, N(0,1) is called a Standard Normal distribution Z-scores are standard normal where

14. Standard Normal Tables Table B in your book gives the percent of the data to the left of the z value. Or in your Standard Normal table Find the 1st 2 digits of the z value in the left column and move over to the column of the third digit and read off the area. To find the cut-off point given the area, find the closest value to the area ‘inside’ the chart. The row gives the first 2 digits and the column give the last digit

15. Calculator Commands To find the percentage of the data between to numbers Z low and Z high : normalcdf(Z low, Z high ) 2nd-VARS==>2:normalcdf

16. Solving a Normal Proportion State the problem in terms of a variable (say x) in the context of the problem Draw a picture and locate the required area Standardize the variable using z =(x-µ)/ Use the calculator/table and the fact that the total area under the curve = 1 to find the desired area. Answer the question.

17. Finding a Cutoff Given the Area State the problem in terms of a variable (say x) and area Draw a picture and shade the area Use the table or the calculator to find the z value with the desired area to the left Go z standard deviations from the mean in the correct direction. Answer the question.