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Introduction to Normal Introduction to Normal Distributions and the Standard Normal Distribution Properties of a Normal Distribution Properties of a Normal Distribution • A A normal distribution is a continuous normal distribution is a continuous probability for a random variable x. • The mean, median and mode are equal The mean median and mode are equal • The normal curve is a bell shaped and is symmetric about the mean i b h • The total area under the curve is equal to one • The normal curve approaches but never touches the x‐axis Normal Curve -3 -2 -1 0 μ‐σ 1 2 μ+σ μ 3 A 0 B Normal Curve 1 2 3 4 5 μ=3.5, σ=1.5 6 7 0 C Normal Curve 1 2 3 4 5 6 7 μ=3.5, σ=0.7 Curve A and Curve B have the same mean, C Curve B and Curve C have the same standard deviation, B dC Ch th t d d d i ti Each Curve has a total area of 1. 0 Normal Curve 1 2 3 4 5 μ=1.5, σ=0.7 6 7 A B Normal Curve 3 6 9 12 15 18 21 Normal Curve 3 6 9 12 15 18 21 • Which normal curve has a greater mean? • Curve A C A • Which normal curve has a greater standard deviation? • Curve B The Standard Normal Distribution The Standard Normal Distribution Standard Normal Curve -3 -2 -1 0 1 2 3 The Standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Properties of the Standard Normal Distribution b • The The cumulative area is close to 0 for z cumulative area is close to 0 for z‐scores scores close to z = ‐3.49 • The cumulative area increases as the z‐scores The cumulative area increases as the z scores increase • The cumulative are for z=0 is 0.500 Th l i f 0 i 0 500 • The cumulative are is close to 1 for z‐scores close to z = 3.49 Using the Standard Normal Table Using the Standard Normal Table Find the cumulative area that corresponds to a z‐score corresponds to a z score of 1.15 of 1.15 Find the cumulative area that corresponds to a z‐score corresponds to a z score of of ‐0.24 0.24 Standard Normal Curve -3 3 -2 -1 0 .8749 8749 1 2 3 Standard Normal Curve -3 3 -2 -1 0 .4052 4052 1 2 3 Finding Area Under the Standard Normal Curve l Find the area under the standard normal curve to the left of z = ‐0.99, then draw and shade the area under the curve. From the standard normal table the area is equal to 0.1611 Standard Normal Curve -3 -2 -1 0 1 2 3 Finding Area Under the Standard Normal Curve l Find the area under the standard normal curve to the left of z = 2.13, then draw and shade the area under the curve. From the standard normal table the area is equal to 0.9834 Standard Normal Curve -3 -2 -1 0 1 2 3 Finding Area Under the Standard Normal Curve Normal Curve Find the area under the standard normal curve to the right of z = 1.06, then draw and shade the area under the curve. From the standard normal table the area is equal to 0.8554, so to the right it is 1 ‐ 0.8554 = 0.1446 Standard Normal Curve -3 -2 -1 0 1 2 3 Finding Area Under the Standard Normal Curve Normal Curve Find the area under the standard normal curve to the right of z = ‐2.16, then draw and shade the area under the curve. From the standard normal table the area is equal to 0.0154, so to the right it is 1 ‐ 0.0154 = 0.9846 Standard Normal Curve -3 -2 -1 0 1 2 3 Finding Area Under the Standard Normal Curve Normal Curve Find the area under the standard normal curve between z = ‐1.5 and z = 1.25, then draw and shade the area under the curve. From the standard normal table the area’s are 0.0668 and 0.8944, so the are between is 0.8944 – 0.0668 = 0.8276 Standard Normal Curve -3 -2 -1 0 1 2 3 Finding Area Under the Standard Normal Curve Normal Curve Find the area under the standard normal curve between z = ‐2.16 and z = ‐1.35, then draw and shade the area under the curve. From the standard normal table the area’s are 0.0154 and 0.0885, so the are between is 0.0885 – 0.0154 = 0.0731 Standard Normal Curve -3 -2 -1 0 1 2 3