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Chapter 3: Normal Distribution

Density curvesNormal distributionsThe 68-95-99.7 ruleFinding the normal proportions

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Distribution of a Continuous Var.

It is described by a density curve The density curve for a continuous var. X is

a curve such that

Proportion of X in between a and b is

the area under it over the interval [a, b]The properties of a density curve:

– It is always on or above the horizontal axis– The total area underneath it is one

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Normal Distribution

The “model” distribution of a continuous var. The normal density curve looks like:

The standard normal density curve centers at 0 with standard deviation 1

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Finding Normal Proportions

X: a normal var. with mean and standard deviation

P(X < a)

= P( )

= see Table A (p. 690-691)

aX

z score

Finding Normal Proportions

1. State the problem and draw a picture

2. Calculate z scores and mark them on the picture

3. Use Table A

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Example

Suppose that the final scores of STAT1000 students follow a normal distribution with = 70 and = 10. What is the probability that a ST1000 student has final score 85 or above (grade A)?

Between 75 and 85 (grade B)? Below 50 (F)?

Finding a Value given a Proportion

3 Steps:

1. State the problem and draw a Z curve picture

2. Use the table to find the z score

3. Unstandardize: find the corresponding x score

Example: What is the first quartile of STAT 100 final score?

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(for Bell-shaped distributions only)

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Empirical Rule (68-95-99.7 rule)

If a variable X follows normal distribution, that is, all X values (the whole population) show bell-shaped, then:

Mean(X) + 1*SD(X) covers 68% of possible X values

Mean(X) + 2*SD(X) covers 95% of possible X values

Mean(X) + 3*SD(X) covers 99.7% of possible X values

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Empirical Rule (68-95-99.7 rule)

If the data (from a sample) of a variable X show bell-shaped, then:

X + 1*S covers about 68% of possible X values

X + 2*S covers about 95% of possible X values

X + 3*S covers about 99.7% of possible X values

How to use Empirical Rule

Find the range covering 68%, 95% or 99.7% of X values

Check if X follows a normal distribution.

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