Chapter Six Normal Curves and Sampling Probability Distributions

Chapter 6 Section 2 Standard Units and Areas Under the Standard Normal Distribution

Z Score The z value or z score tells the number of standard deviations the original measurement is from the mean. The z value is in standard units.

Formula for z score

Calculating z-scores The amount of time it takes for a pizza delivery is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. Convert 21 minutes to a z-score.

Calculating z-scores Mean delivery time = 25 minutes Standard deviation = 2 minutes Convert 29.7 minutes to a z score.

Interpreting z-scores Mean delivery time = 25 minutes Standard deviation = 2 minutes Interpret a z score of 1.6. The delivery time is 28.2 minutes.

Standard Normal Distribution: μ = 0 σ = 1 Values are converted to z scores where

Importance of the Standard Normal Distribution: 1 0 11  The areas are equal. Any Normal Distribution: Standard Normal Distribution:

Use of the Normal Probability Table (Table 4) - Appendix I Entries give the probability that a standard normally distributed random variable will assume a value between the mean (zero) and a given z-score.

Z-Scores z To find the area between z = 0 and z = 1.34

Patterns for Finding Areas Under the Standard Normal Curve To find the area between a given z value and zero: Use Table 4 (Appendix I) directly. z 0

Patterns for Finding Areas Under the Standard Normal Curve To find the area between z values on either side of zero: Add area from z 1 to zero to area from zero to z 2. z2z2 0 z1z1

Patterns for Finding Areas Under the Standard Normal Curve To find the area between z values on the same side of zero: Subtract area from zero to z 1 from the area from zero to z 2. z2z2 0 z1z1

Patterns for Finding Areas Under the Standard Normal Curve To find the area to the right of a positive z value or to the left of a negative z value: Subtract the area from zero to z from z

Patterns for Finding Areas Under the Standard Normal Curve To find the area to the left of a positive z value or to the right of a negative z value: Add to the area from zero to z. z table

Use of the Normal Probability Table a. P(0 < z < 1.24) = _________________ b. P(0 < z < 1.60) = _________________ c. P( < z < 0) = ________________

Normal Probability d. P( - 3 < z < 3 ) = ____________________ e. P( < z < 1.57 ) = ______________ f. P( 1.24 < z < 1.88 ) = ________________ g. P(-3.52<z< -0.98) = __________________

h. P(z < 1.64) = _________________ i. P(z > 2.39) = __________________ j. P(z > -1.35) = _________________ k. P(z < -0.64) = _________________ Normal Probability

Application of the Normal Curve The amount of time it takes for a pizza delivery is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. If you order a pizza, find the probability that the delivery time will be: a. between 25 and 27 minutes. a. ____________ b. less than 30 minutes. b. ____________ c. less than 22.7 minutes. c. ____________

Application of the Normal Curve The amount of time it takes for a pizza delivery is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. If you order a pizza, find the probability that the delivery time will be: a.between 25 and 27 minutes

Application of the Normal Curve The amount of time it takes for a pizza delivery is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. If you order a pizza, find the probability that the delivery time will be: b.less than 30 minutes

Application of the Normal Curve The amount of time it takes for a pizza delivery is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2 minutes. If you order a pizza, find the probability that the delivery time will be: c.less than 22.7 minutes

Homework Assignments Chapter 6 Section 2 Pages Exercises: , odd Exercises: , even