Continuous Distributions - Statistics - Lecture Slides, Slides of Statistics

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Continuous Distributions

Characteristics of the Normal Distribution

• Continuous distribution
• Symmetrical distribution
• Asymptotic to the

horizontal axis

• Unimodal
• A family of curves
• Area under the curve

sums to 1.

• Area to right of mean is

• Area to left of mean is

μ

1/2 1/

X

Normal Curves for Different

Means and Standard Deviations

σ = 5

σ = 5

σ = 10

Standardized Normal Distribution

• A normal distribution with

– a mean of zero, and

– a standard deviation of one

• Z Formula

– standardizes any normal

distribution

• Z Score

– computed by the Z Formula

– the number of standard

deviations which a value is

away from the mean

Z

X

σ

Table Lookup of a

Standard Normal Probability

P ( 0 ≤ Z ≤ 1 ) = 0 3413.

Z 0.00 0.01 0.

Applying the Z Formula

X is normally distributed with μ = 485, and σ = 105

P ( 485 ≤ X ≤ 600 ) = P ( 0 ≤ Z ≤ 1 10. ) =. 3643

For X = 485,

Z =

X - μ

For X = 600,

Z =

X - μ

Z 0.00 0.01 0.

Normal Approximation of Binomial:

Parameter Conversion

• Conversion equations
• Conversion example:

μ

σ

= ⋅

= ⋅ ⋅

n p

n p q

Given that X has a binomial distribution, find

P X n and p

n p

n p q

( |. ).

( )(. )

( )(. )(. ).

≥ = =

= ⋅ = =

= ⋅ ⋅ = =

25 60 30

60 30 18

60 30 70 3 55

μ

σ

Normal Approximation of

Binomial: Interval Check

n

Normal Approximation of

Binomial: Graphs

Normal Approximation of Binomial:

Computations

Total

X P(X)

The normal approximation,

P(X ≥ 24.5| = and =

P Z

P Z

P Z