Chapter 6 Part 1 [statistic For I.t]

Chapter 6 Part 1 The Normal Distribution

Normally Distributed Variable

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Standard Normal Distribution; Standard Normal Curve • A normally distributed variable having mean 0 and standard deviation 1 is said to have the standard normal distribution. Its associated normal curve is called standard normal curve, which is shown below:

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Standardized Normally Distributed Variable • We standardize a variable x by subtracting its mean and then dividing by its standard deviation.

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Standardized Normally Distributed Variable •Theoretically, it says that standardizing converts all normal distributions to the standard normal distribution.

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Standardized Normally Distributed Variable Practically, let x be a normally distributed variable with mean μ and standard deviation σ, and let a and b be real numbers with a < b. The percentage of all possible observations of x that lie between a and b is the same as the percentage of all possible observations of z that lie between (a −μ)/σ and (b−μ)/σ. This latter percentage equals the area under the standard normal curve between (a −μ)/σ and (b−μ)/σ.

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Basic Properties of the Standard Normal Curve

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Example: Finding Area • Using Table II (in Statistical Tables) • Finding the Area to the left of a specified z-Score • To the left of z = 1.23 • Finding the Area to the right of a specified z-Score • To the right of z = 0.76 • Finding the Area to the between two specified z-Scores • Between z = -0.68 and z = 1.82

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Example: Finding the z-score @ Zα • Using Table II (in Statistical Tables) • Finding the z-Score having a Specified Area to its left. • Determine the z-score having an area of 0.04 to its left under the standard normal curve.

Use the closest to 0.04, which is 0.0401. The corresponding z.score is -1.75

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Example: Finding the z-score @ Zα • •

Use Table II (in Statistical Tables) Find

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Z 0.025

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Z 0.05

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