Hypothesis testing and Decision Making

Formal aspects of hypothesis testing

Null and Alternative Hypotheses Null hypothesis (H0)

sets the ‘what if’ for calculating probabilities

Alternative hypothesis (Ha) sets the rejection region. Oddly enough, the alternative is usually what you have in mind to prove during the study.

50:0 H

50: aH

0: 210 H

0: 21 aH

TailsThe rejection region can fit into 1 or 2 tails of the sampling distribution of means. The RR is determined by the alternative hypothesis.

Two Tails One Tail

valueH :0

valueH a :

valueH :0

orvalueH a :

valueH a :

Tails illustratedvalueH a :

3210-1-2-3Z

Z

Z

1.96-1.96

Don't reject RejectReject

Likely Outcome If Null is True

3210-1-2-3Z

Z

Don't reject Reject

Likely Outcome If Null is True

1.65

Two tails.

One tail.

valueH a :

Note 1.96 vs. 1.65

Example of 2 tails

Suppose:

Then:

75:;75:0 aHH25,10 N

92.7808.7125

1096.175

X

78.9271.08

Don't reject RejectReject

Likely Outcome If Null is True

75

Note 5 percent is split into two tails.

Example of 1 tail

Suppose:75:;75:0 aHH

25,10 N

3.7825

1065.175

8079787776757473727170

X

0.20

0.16

0.12

0.08

0.04

0.00

8079787776757473727170

X

Sampling Distribution of Means

Likely Outcome if Null is True

Don't RejectReject

78.3Note all 5 percent is at the top tail.

Review

Suppose Draw sampling

distribution of means. What is the shape of this

distribution? What is the mean of this

distribution? What is the standard

deviation of this distribution?

Draw RR if

100;100;650:0 NH

650: aH650: aH

680670660650640630620

Sampling Distribution of Means

Statistical Decision Making

Must decide how to act even if uncertain (developed by Fisher for agriculture)

Make decisions like bets in gambling (null is false if null is improbable)

If we do this, some times we will be right; sometimes wrong.

We can calculate probabilities of mistakes and correct decisions. Some have names.

Decisions

Null true

Null False

Accept Null

Right, but…

Beta (type II error)

Reject

Null

Alpha

(type I error)

Correct rejection (power)

Population Condition

Sample Decision

Fire alarm No fire Fire

Alarm silent

Right, but…

Beta

Alarm sounds

Alpha Correct rejection

Three named probabilities:Alpha, beta, and power.

Researcher’s Choice

We can pick alpha (but usually .05) We can improve power by

Good alternative hypothesis Good design (minimize error) Large samples

Review

Define the following Alpha Beta Power

Why is it good to have alpha be a small number?

Why is it good to have power be a large number?

Definition

The alternative hypothesis sets the placement of the _____. 1 alpha 2 omega 3 rejection region 4 standard error

Definition

Alpha refers to what kind of error? 1 descriptive 2 primary 3 type I 4 type II

Application

A researcher can increase the _____ to increase the power of a study. 1 number of outcomes 2 sample size 3 standard error 4 study duration