1. Standard Normal Calculations 2.2 Standard Normal Calculations 2.2 Target Goal: I can standardize individual values and compare them using a common scale

2. Standardizing and Z-Scores All normal curves are the same if we measure in units of size σ about the mean μ as center. Changing to these units is called standardizing.  If x is an observation from a distribution that has mean μ and standard deviation σ, then the standardized value, called the z-score of x is:

3. Standardizing and Z-Scores A z-score tells us how many standard deviations the original observation falls away from the mean and in which direction.  Observations larger than the mean are positive.  Observations smaller than the mean are negative.

4. Standard Normal Distribution  If a variable x has any normal dist. N(μ,σ),  Then the new standardized variable produced has the standard normal distribution. Original Standardize New x, N(μ,σ) z z, N(0,1)

5. Ex 1: Standardizing Women’s Heights  The heights of young women are approx. normal with μ = 64.5 inches and σ = 2.5 inches.  The standardized height is: z = (height – )/ z = (height – )/ A women’s standardized height is the number of standard deviations by which her height differs from the mean height of all young women. 64.52.5

6. μ = inches and σ = μ = 64.5 inches and σ = 2.5  A women 68 inches has a standardized height ? z = ( – )/ = standard deviations above the mean.  A women 5 feet tall has a standardized height? standard deviations below the mean. standard deviations below the mean. 68 64.52.5 z = (60 – 64.5)/2.5 =-1.8 1.4

7. Standardizing  Standardizing gives a common scale and produces a new variable that has the standard normal distribution.

8. Normal Distribution Calculations  The area under a density curve is a proportion of the observations in a distribution.  After standardizing, all normal distributions are the same.  Table A gives areas under the curve for the standard normal distribution.

9. The table entry for each z value is the area under the curve to the left of z.  Be careful if the problem asks for the area to the left or to the right of the z value.  Always sketch the normal curve, mark the z value, and shade the area of interest.

10. Ex. 3 Using the Z Table  Find the proportion of observations from the standard normal distribution that are less than 1.4.  Look in Table A to verify

11. Ex 1. Cont. Women’s Height  What proportion of all young women are less than 68 inches tall. (We already calculated z).  The area to the left of z = 1.4 under the standard normal curve is the same as the area to the left of x = 68. (Use Table A)

12. Finding Normal Proportions 1.State: the problem in terms of the observed variable x. Draw a picture of the distribution and shade the area of interest under the curve. 2.Plan: Standardize x and restate the problem in terms of a standard normal variable z. Draw a picture to show the area of interest under the standard normal curve. 3.DO: Find the required area under the standard normal curve, using Table A and the fact that the total area under the curve is 1. 4.Conclude: Write your conclusion in the context of the problem.

13. Exercise 4: Table A Practice  Use Table A. In each case, sketch a standard normal curve and shade the area under the curve that is the answer to the question. a.z < 2.85 b.z > 2.85 c.z > -1.66 d.-1.66 < z < 2.85.9978 1-.9978 =.0022 1 -.0485 =.9515.9978 -.0485 =.9493