6.2 Area Under the Standard Normal Curve
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1 6.2 Area Under the Standard Normal Curve Tom Lewis Fall Term 2009 Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
2 Outline 1 The cumulative distribution function 2 The z α notation Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
3 Cumulative distribution Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
4 Cumulative distribution For z (, ), let Φ(z) denote the area under the standard normal curve over the region from to z. Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
5 Cumulative distribution For z (, ), let Φ(z) denote the area under the standard normal curve over the region from to z. Thus Φ(z) represents the probability that a standard normal random variable takes on a value less than z. Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
6 Cumulative distribution For z (, ), let Φ(z) denote the area under the standard normal curve over the region from to z. Thus Φ(z) represents the probability that a standard normal random variable takes on a value less than z. Values for Φ have been tabulated; see Table II. Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
7 Cumulative distribution For z (, ), let Φ(z) denote the area under the standard normal curve over the region from to z. Thus Φ(z) represents the probability that a standard normal random variable takes on a value less than z. Values for Φ have been tabulated; see Table II. R will calculate values of Φ through the pnorm() function. Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
8 Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
9 Find the area under the standard normal curve from to 1.31 Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
10 Find the area under the standard normal curve from to 1.31 Find the probability that a standard normal variable has a value less than.24 Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
11 Find the area under the standard normal curve from to 1.31 Find the probability that a standard normal variable has a value less than.24 What is the probability that a standard normal variable assumes a value between 1.32 and 2.01? Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
12 Find the area under the standard normal curve from to 1.31 Find the probability that a standard normal variable has a value less than.24 What is the probability that a standard normal variable assumes a value between 1.32 and 2.01? What is the probability that a standard normal random variable exceeds Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
13 Find the area under the standard normal curve from to 1.31 Find the probability that a standard normal variable has a value less than.24 What is the probability that a standard normal variable assumes a value between 1.32 and 2.01? What is the probability that a standard normal random variable exceeds If the probability that a standard normal random variable is greater than z is.18, then what is z? Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
14 The z α notation The z α notation The symbol z α is used to denote the z-score that has area α to its right under the standard normal curve. In other words, 1 Φ(z α ) = α or Φ(z α ) = 1 α. Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
15 The z α notation The z α notation The symbol z α is used to denote the z-score that has area α to its right under the standard normal curve. In other words, 1 Φ(z α ) = α or Φ(z α ) = 1 α. Using R The R function qnorm() can be used to calculate z α values. Given an number 0 < p < 1, qnorm(p) calculates the z value such that the area under the standard normal curve from up to z is p. Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
16 The z α notation Use Table II and R in the following problems: Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
17 The z α notation Use Table II and R in the following problems: Find z.05 Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
18 The z α notation Use Table II and R in the following problems: Find z.05 Find z.65 Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
19 The z α notation Use Table II and R in the following problems: Find z.05 Find z.65 Find the value a > 0 such that the area trapped under the standard normal curve over the symmetric interval [ a, a] is.88. Tom Lewis () 6.2 Area Under the Standard Normal Curve Fall Term / 6
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