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Chapter 7 Section 2 The Standard Normal Distribution

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Chapter 7 – Section 2 Learning objectives Find the area under the standard normal curve Find Z-scores for a given area Interpret the area under the standard normal curve as a probability 1 2 3

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Chapter 7 – Section 2 Learning objectives Find the area under the standard normal curve Find Z-scores for a given area Interpret the area under the standard normal curve as a probability 1 2 3

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Chapter 7 – Section 2 The standard normal curve is the one with mean μ = 0 and standard deviation σ = 1 We have related the general normal random variable to the standard normal random variable through the Z-score In this section, we discuss how to compute with the standard normal random variable

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Chapter 7 – Section 2 There are several ways to calculate the area under the standard normal curve What does not work – some kind of a simple formula We can use a table (such as Table IV on the inside back cover) We can use technology (a calculator or software) Using technology is preferred

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Chapter 7 – Section 2 Three different area calculations Find the area to the left of Find the area to the right of Find the area between Three different area calculations Find the area to the left of Find the area to the right of Find the area between Three different methods shown here From a table Using Excel Using Statistical software

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"To the left of" – using a table Calculate the area to the left of Z = 1.68 Break up 1.68 as 1.6 +.08 Find the row 1.6 Find the column.08 Read answer at intersection of the two. The probability is 0.9535 Enter Read Enter

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The blue area is 0.9535 Here is what is looks like on the normal distribution curve.

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To find area greater than 1.68 is 1 –.9535 =.0465 The red area is 0.0465 The blue area is 0.9535

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Chapter 7 – Section 2 "To the right of" – using a table The area to the left of Z = 1.68 is 0.9535 Read Enter The area to the left of Z = 1.68 is 0.9535 The right of … thats the remaining amount The two add up to 1, so the right of is 1 – 0.9535 = 0.0465

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Chapter 7 – Section 2 Between Between Z = – 0.51 and Z = 1.87 This is not a one step calculation

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Chapter 7 – Section 2 The left hand picture … to the left of 1.87 … includes too much It is too much by the right hand picture … to the left of -0.51 Included too much Included too much

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Chapter 7 – Section 2 Between Z = – 0.51 and Z = 1.87 We want We start out with, but its too much We correct by

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Chapter 7 – Section 2 Between Z = – 0.51 and Z = 1.87 This area for 1.87 is 0.9693 This area for -.51 Is 0.3050. 9693-.3050=.6643

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We can use any of the three methods to compute the normal probabilities to get: The area to the left is read directly from the chart The area to the right of 1.87 is 1 minus area to the left. This area for -.51 Is 0.3050 Area left of 1.87 is 0.9693 so area to the right is 1- 0.9693

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The area between -0.51 and 1.87 The area to the left of 1.87, or 0.9693 … minus The area to the left of -0.51, or 0.3050 … which equals The difference of 0.6643 Thus the area under the standard normal curve between -0.51 and 1.87 is 0.6643. 9693-.3050=.6643

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We can use any of the three methods to compute the normal probabilities to get: The area to the left is read directly from the chart The area to the right of 1.87 is 1 minus area to the left. The area between -0.51 and 1.87 The area to the left of 1.87= 0.9693 Minus area to the left of -0.51= 0.3050 Which equals the difference of 0.6643 This area for -.51 Is 0.3050 Area left of 1.87 is 0.9693 so area to the right is 1- 0.9693. 9693-.3050=.6643

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Chapter 7 – Section 2 Learning objectives Find the area under the standard normal curve Find Z-scores for a given area Interpret the area under the standard normal curve as a probability 1 2 3

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Chapter 7 – Section 2 We did the problem: Z-Score Area Now we will do the reverse of that Area Z-Score We did the problem: Z-Score Area Now we will do the reverse of that Area Z-Score This is finding the Z-score (value) that corresponds to a specified area (percentile) And … no surprise … we can do this with a table, with Excel, with StatCrunch, with …

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Chapter 7 – Section 2 To the left of – using a table Find the Z-score for which the area to the left of it is 0.32 Look in the middle of the table … find 0.32 Find Read Find the Z-score for which the area to the left of it is 0.32 Look in the middle of the table … find 0.32 The nearest to 0.32 is 0.3192 … a Z-Score of -.47

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Chapter 7 – Section 2 "To the right of" – using a table Find the Z-score for which the area to the right of it is 0.4332 Right of it is.4332 … left of it would be.5668 A value of.17 Enter Read

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We will often want to find a middle range, to find the middle 90% or the middle 95% or the middle 99%, of the standard normal The middle 90% would be

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Chapter 7 – Section 2 90% in the middle is 10% outside the middle, i.e. 5% off each end These problems can be solved in either of two equivalent ways We could find The number for which 5% is to the left, or The number for which 5% is to the right

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The two possible ways The number for which 5% is to the left, or The number for which 5% is to the right 5% is to the left5% is to the right

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Chapter 7 – Section 2 The number z α is the Z-score such that the area to the right of z α is α Some useful values are z.10 = 1.28, the area between -1.28 and 1.28 is 0.80 z.05 = 1.64, the area between -1.64 and 1.64 is 0.90 z.025 = 1.96, the area between -1.96 and 1.96 is 0.95 z.01 = 2.33, the area between -2.33 and 2.33 is 0.98 z.005 = 2.58, the area between -2.58 and 2.58 is 0.99

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Chapter 7 – Section 2 Learning objectives Find the area under the standard normal curve Find Z-scores for a given area Interpret the area under the standard normal curve as a probability 1 2 3

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The area under a normal curve can be interpreted as a probability The standard normal curve can be interpreted as a probability density function The area under a normal curve can be interpreted as a probability The standard normal curve can be interpreted as a probability density function We will use Z to represent a standard normal random variable, so it has probabilities such as P(a < Z < b) The probability between two numbers P(Z < a) The probability less than a number P(Z > a) The probability greater than a number

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Summary: Chapter 7 – Section 2 Calculations for the standard normal curve can be done using tables or using technology One can calculate the area under the standard normal curve, to the left of or to the right of each Z-score One can calculate the Z-score so that the area to the left of it or to the right of it is a certain value Areas and probabilities are two different representations of the same concept

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Were are done.

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