7.2 Standard Normal Distribution

1. 7.2 Standard Normal Distribution Obj: Find the area under the standard normal curve and use area to find Z-scores

2. Review Normal Probability Distribution Elena conducts an experiment in which she fills up the gas tank on her Toyota Camry 40 times and records the miles per gallon for each fill-up. A histogram of the miles per gallon indicates that a normal distribution with mean of 24.6 miles per gallon and a standard deviation of 3.2 miles per gallon could be used to model the gas mileage for her car. Draw and label the curve. Shade the region that represents mileage greater than 26. The area shaded is 0.3309. Provide two interpretations of this area.

3. Standard Normal Curve Properties 1. It is symmetric about its mean, 0, and has standard deviation of 1. 2. Its highest point is when μ = 0. 3. Its inflections points are at -1 and 1. 4. The area under the curve is 1. 5. As z increases or decreases, the graph approaches, butnever equals 0. 6. Empirical Rule

4. -3 -2 -1 0 3 2 1 Finding the Area under the Standard Normal Curve When finding area under a normal curve, always sketch a normal curve and shade the area you are finding. Find the area under the standard normal curve that lies to the left of Z = -1.39. -1.39

5. Finding Area • Use the table of z-scores at the back of the book • Use the calculator DISTR normalcdf(lower bound, upper bound, 0, 1) Both always finds area TO THE LEFT of the z-score.

6. Practice Determine the area under the standard normal curve that lies to the left of Z = -2.45 left of Z = 3.49 right of Z = -2.45 right of Z = 2.91 between Z = -2.04 and Z = 2.04 between Z = -0.55 and Z = 0

7. Finding a Z-score given an area Working backward Find the Z-score so that the area to the left of the Z-score is 0.57. Table Calculator DISTR invnorm(area left, 0, 1)

8. Assignment Page 381 6, 8, 10, 11 – 43 odd