The Normal Distributions and Sampling Distributions - Sections 5.5, 6.5 & 6

The Normal Distributions and Sampling Distributions

Sections 5.5, 6.5 & 6.6

Cathy Poliak, Ph.D.

cathy@math.uh.edu

Office in Fleming 11c

Department of Mathematics

University of Houston

Lecture 8 - 3339

Cathy Poliak, Ph.D. cathy@math.uh.edu Office in FlemingSections

11c (Department

5.5, 6.5 & of

6.6Mathematics University of Houston

Lecture )8 - 3339

1 / 78

Outline

1

Normal Distribution

2

The Empirical Rule

3

The Standard Normal Distribution

4

Using the z-table

5

Inverse Normal

6

Approximating the Binomial Distribution

7

Sums of Random Variables

8

Sampling Distributions

9

Sampling Distribution of X?

10

Finding Probabilities for X?

11

Proportions

12

Sampling Distribution of p?

Cathy Poliak, Ph.D. cathy@math.uh.edu Office in FlemingSections

11c (Department

5.5, 6.5 & of

6.6Mathematics University of Houston

Lecture )8 - 3339

2 / 78

Definition of a Density Function

A density function is a nonnegative function f defined of the set

of real numbers such that:

Z ��

f (x)dx = 1.

?��

Rx

If f is a density function, then its integral F (x) = ?�� f (u)du is a

continuous cumulative distribution function (cdf), that is

P(X �� x) = F (x).

If X is a random variable with this density function, then for any

two real numbers, a and b

Z b

P(a �� X �� b) =

f (x)dx.

a

Cathy Poliak, Ph.D. cathy@math.uh.edu Office in FlemingSections

11c (Department

5.5, 6.5 & of

6.6Mathematics University of Houston

Lecture )8 - 3339

3 / 78

Using the cdf F (X ) to Compute Probabilities

Let X be a continuous random variable with pdf f (x) and cdf F (x).

Then for any number a,

P(X > a) = 1 ? F (a)

and for any two numbers a and b with a < b,

P(a �� X �� b) = F (b) ? F (a)

Cathy Poliak, Ph.D. cathy@math.uh.edu Office in FlemingSections

11c (Department

5.5, 6.5 & of

6.6Mathematics University of Houston

Lecture )8 - 3339

4 / 78

The Normal distributions

Common type of probability distributions for continuous random

variables.

The highest probability is where the values are centered around

the mean. Then the probability declines the further from the mean

a value gets.

These curves are symmetric, single-peaked, and bell-shaped.

The mean ? is located at the center of the curve and is the same

as the median.

The standard deviation �� controls the spread of the curve.

If �� is small then the curve is tall and slim.

If �� is large then the curve is short and fat.

Cathy Poliak, Ph.D. cathy@math.uh.edu Office in FlemingSections

11c (Department

5.5, 6.5 & of

6.6Mathematics University of Houston

Lecture )8 - 3339

5 / 78

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