a grudge is a heavy thing to carry.. probability the probability of an event is the proportion of...

A grudge is a heavy

thing to carry.

Probability

The probability of an event is the proportion of times we would expect the event to occur in an infinitely long

series of identical sampling experiments.

Probability

If all the possible outcomes are equally likely, the

probability of the occurrence of an event is equal to the proportion of the possible outcomes characterized by

the event.

Probability

Probability is a very useful notion in situations involving

at least some degree of uncertainty; it gives us a way of expressing the degree of assurance that a particular

event will occur.

Probability

Chance factors inherent in forming samples always affect sample results. Sample results must therefore be interpreted

with that in mind.

Probability

The probability of occurrence of either one event OR another

OR… is obtained by adding their individual probabilities, provided

the events are mutually exclusive.

Probability

The probability of the joint occurrence of one event AND

another AND another AND another… is obtained by

multiplying their separate probabilities, provided the events are independent.

Probability

Any relative frequency distribution may be interpreted

as a probability distribution.

Probability

Pr = .15 + .30 + .40 = .85 (85% chance)

Grade Relative Frequency

A .15

B .30

C .40

D .10

F .05

Your CCHHAANNCCEEFor a break coming up soon!

Confidence Statement

Statistic ± Margin of Error

Margin of Error

1

n

Confidence Interval

A range of values constructed from sample data so the

parameter occurs within that range at a specified

probability. The specified level of probability is called

the level of confidence.

Confidence Interval for a Sample Mean

X + Z((

SN

We’re We’re CONFIDENTCONFIDENT

there’s a there’s a P P R R O O B B A A B B I I L L I I T T YY

you want a breakyou want a break

Stand up and take

a short break

Confidence Interval

95% of those surveyed will fall

into a certain range surrounding the mean

95% Confidence Interval

Confidence Interval

The average size of a mortgage applied for in 1993 was $116,991 as

opposed to $119,999 in 1992. A sample of 64 mortgages showed that the standard deviation of the amount applied for was $6019. Find a 95% confidence interval for the average

size of a mortgage applied for in 1993.

Z = 1.96

Confidence Interval

s = 6019

C =95%i

n = 64

x = $116,991

Confidence Interval

X ± Z( s / √n) = 116,991 ± 1.96( 6019 / √64 ) =

116,991 ± 1.96(752.38) =116,991 ± 1474.66 =

115516.34 to

118465.66

Confidence Interval

According to the Family Economic Research Group of the US Department of Agriculture, middle income couples who had babies in 1992 will spend an average of $128,670 by the time the baby is 18 years old. Assume the standard deviation of a sample of 100

families was $8473. Find a 90% confidence interval for the average

cost to raise a child born in 1992.

Z = 1.645

n = 100s = 8473

X = 128,670C = 90%

i

Confidence Interval

Confidence Interval

X ± Z( s / √n) =

128,670 ± 1.645( 8473 / √100) =

128,670 ± 1.645(847.3) =

128670 ± 1393.81 =

127276.19

to

130063.81

Confidence Interval for Confidence Interval for aa Sample Proportion Sample Proportion

p + Z p(1 - p(

n

Confidence Interval for a Sample Proportion

Suppose 1,600 of 2000 union members sampled said they plan to vote for the

proposal to merge with the UMA. Using the .95 level of confidence, what is the

interval estimate for the population proportion? Based on the

confidence interval,what conclusion can be drawn?

p + Z p (1 - p

(n

=.80 + 1.96.80(1-.80)

2,000

= .80 + 1.96 .00008

= .782 and .818

Confidence Interval for a Sample Proportion

Point Estimate

A value, computed from sample information that is used to

estimate the population parameter

Standard Error of the Sample Mean

The standard deviation of the sampling distribution of the

sample means. It is a measure of the variability of the sampling distribution of

the sample mean.

Look at the explanation provided in your textbook

Pages 429-434

Central Limit Theorem

REMINDER:REMINDER:

Chapter 18 will be an important reference

for this section of statistics

Experimental Process

Subjects

Treatment

Observation

VariablesExplanatory Variable

(Independent variable)

Response Variable(Dependent variable)

Lurking of Confounding Variable

Alternative Experimental Designs

Completely Randomized Design

Block Design

Matched Pairs Design

Double Blind Design

Completely Randomized Design

Simplest Design Strategy

Each subject is randomly assigned to one group

Typically, group sizes are

identical

Completely Randomized Design SubjectsSubjects Group AGroup A Group BGroup B Group CGroup C

AdamsAdams XX

AllenAllen XX

BaileyBailey XX

DaltonDalton XX

GrayGray XX

JamesJames XX

RobertsRoberts XX

SmithSmith XX

WhiteWhite XX

Block DesignUsed when known extraneous variables

may influence the experiment

Subjects are pre-sorted by the influencing variables, then partitioned into similar

blocks

Subjects from each block are randomly

assigned to groups

One-Dimensional Block Design

to Control AgeAge Subject Group A Group B Group C

16 Gray

17 May May Gray Lee

20 Lee

28 Jones

29 Cooper Smith Jones Cooper

29 Smith

30 Adams

33 Brown Brown Adams Magee

34 Magee

Matched Pairs Design

Each subject receives each treatment

Treatment sequence is randomly chosen for each

subject

Matched Pairs Design

Subjects Treatment Order

Adams A B C

Allen A C B

Bailey B A C

Dalton C B A

Gray A B C

James C B A

Roberts B C A

Smith C A B

White A C B

Double Blind Experiment

Neither the subjects nor the investigators know which treatment is administered

Control

Minimize the effects of lurking/confounding variables on the response, most simply

by comparing several treatments.

Randomize

Use impersonal chance to assign subjects to treatments.

Replicate

Repeat the experiment on many subjects to reduce chance

variation in the results.

Statistical Significance

An observed effect so large that it would rarely occur by chance.