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Chapter 6: Q. 6.87 (page 276)

Explain why the percentage of all possible observations of a normally distributed variable that lie within two standard deviations to either side of the mean equals the area under the standard normal curve between -2and2.

Short Answer

Expert verified

The area under the standard normal curve for the range -2 to 2 is then the percentage of observations within the range.

Step by step solution

01

Given information

For a normally distributed random variable, the proportion of all possible values within the interval of two standard deviations from either side of the mean is equal to the area between -2 and 2 under the standard normal curve.

02

Explanation

The random variable Xhave z-score which is given by

z=x-μσ

Where μis the mean and

σis the standard deviation

Let the interval limits be

a=μ-2σ

b=μ+2σ

For a,

z=(μ-2σ)-μσ

=-2.

For b,

z=(μ+2σ)-μσ

=2

Hence the values that lie at two standard deviations from the mean have the z-scores-2or2.

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Most popular questions from this chapter

Obtain the following z-scores

a. z0.20

b.z0.06

Find the z-score that has an area of 0.75 to its left under the standard normal curve.

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