the language of statistical decision making

The Languageof Statistical

DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel

IncreasedSample Size

Assignment

Answers toEven-NumberedProblems

The Language of Statistical Decision MakingLecture 3

Section 1.3

Robb T. Koether

Hampden-Sydney College

Mon, Aug 31, 2009


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Outline

1 Homework Review

2 Two Types of Error

3 Significance Level

4 Increased Sample Size

5 Assignment

6 Answers to Even-Numbered Problems


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Homework Review

Exercise 1.3, page 67.

The article states that of those receiving other treatments,10% survive five years. State the appropriate hypothesesfor assessing whether this vaccine is just as effective asother treatments, or if this vaccine is more effective thanother treatments, in terms of the five-year survival rates.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Homework Review

SolutionThe hypotheses are

H0 : The 5-year survival rate for the new vaccine is 10%.H1 : The 5-year survival rate for the new vaccine isgreater than 10%.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Study: Fear of Snakes and Spiders

ArticleGirls Have Head Start on Snake and Spider Fears

http://www.sciencenews.org/view/generic/id/46784/title/Girls_have_head_start_on_snake_and_spider_fears


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Errors

The hypotheses wereH0 : Infant girls and infant boys have the same fear ofsnakes and spiders.H1 : Infant girls have a greater fear of snakes andspiders than infant boys do.

Based on the evidence, the null hypothesis wasrejected in favor of the alternative hypothesis.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Errors

Do we now know that infant girls have a greater fear ofsnakes and spiders than infant boys do?If not, why not?


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Errors

Had the study concluded that there was no difference ininfant girls’ and infant boys’ fear of snakes and spiders,would we then know that there was no difference?If not, why not?


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Two Types of Error

Definition (Type I Error)A Type I error is to reject the null hypothesis when it is true,i.e., when it should have been accepted.

Definition (Type II Error)A Type II error is to accept the null hypothesis when it isfalse, i.e., when it should have been rejected.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Decisions and Errors

H0 is true H0 is false

Accept H0

Reject H0

The State of Nature

Our Decision

Correct

Correct

Type IError

Type IIError


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment



Girls, boysequallyscared

Girls morescared than

boys

The State of Nature

Our Decision

Correct

Correct

Type IError

Type IIError

Girls, boysequallyscared

Girls morescared than

boys


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment



H0 is true H0 is false

Accept H0

Reject H0

The State of Nature

Our Decision

Correct

Correct

Type IError

Type IIError


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Types of Error

Example (Types of error)A person is tested for H1N1 infection.There are two possible realities (hypotheses):

The person is infected with H1N1.The person is not infected with H1N1.

Which one should be the null hypothesis?


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Types of Error

Example (Types of error)There are two possible diagnoses (conclusions):

The person is infected with H1N1.The person is not infected with H1N1.

Describe a Type I error.Describe a Type II error.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Types of Error

Example (Types of error)

Not infectedwith H1N1

Infectedwith H1N1

Not infectedwith H1N1

Infectedwith H1N1

The State of Nature

Our Decision

Correctdiagnosis

Correctdiagnosis

Falsepositive

Falsenegative


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Statistical Significance

DefinitionThe data are called statistically significant if their deviationfrom what would be expected under the null hypothesis istoo great to be attributed to sampling error (chance).


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Significance Level

Definition (Significance Level)

The significance level of a test is the likelihood of rejectingH0 when it is true, i.e., the likelihood of committing a Type Ierror.

The symbol α represents the likelihood of a Type I error.The symbol β represents the likelihood of a Type IIerror.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Significance Level

Example (Significance level)

Suppose that we have two unusual dice.Die A rolls a 1 80% of the time and a 4 only 20% of thetime.Die B rolls a 1 only 10% of the time and a 4 90% of thetime.

Outcome Die A Die B1 0.80 0.104 0.20 0.90

Visually, the two dice are indistinguishable.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Significance Level


We pick up one of the dice.Let the hypotheses be

H0: We picked up die A.H1: We picked up die B.

We will roll the die one time and, based on theoutcome, decide which die we think it is.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Significance Level


What should be our decision rule for choosing betweenthe two hypotheses?That is, if the die turns up 1, which hypothesis do wechoose?What if it turns up 4?Describe a Type I error.Describe a Type II error.What is the value of α?What is the value of β?


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Increased Sample Size

It seems that more evidence should lead to a morereliable conclusion.Can we reduce the probability of error by getting alarger sample?


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment



Example (Increased sample size)Suppose now that we roll the chosen die three timesand use the average of the three rolls to make ourdecision.We must get either

Three 1s, with an average of 1.Two 1s and one 4, with an average of 2.One 1 and two 4s, with an average of 3.Three 4s, with an average of 4.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment



Example (Increased sample size)The following table summarizes the situation.

Outcome Average Die A Die BThree 1s 1 0.512 0.001Two 1s, one 4 2 0.384 0.027One 1, two 4s 3 0.096 0.243Three 4s 4 0.008 0.729


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment



Example (Increased sample size)What would be a good decision rule for deciding whichdie it is?Based on this decision rule,

What is α?What is β?


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Assignment

HomeworkRead Section 1.3.3, pages 11 - 15.Let’s Do It! 1.5, 1.6.Page 67, exercises 4 - 12.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Answers to Even-Numbered Problems

Page 671.4 H0 : The new technique is not effective (40% of the whales

leave).H1 : The new technique is effective (more than 40% of the

whales leave).1.6 (a) H0.

(b) H0.(c) H0.(d) H1.

1.8 (a) A Type I error is to conclude that the employee is a druguser when, in fact, the employee is not a drug user. AType II error is to conclude that the employee is not adrug user when, in fact, the employee is a drug user.

(b) 0.04.


DecisionMaking

Robb T.Koether

HomeworkReview

Two Types ofError

SignificanceLevel


Assignment


Answers to Even-Numbered Problems

Page 67

1.10 You could use the article about girls and boys and fearof snakes and spiders.

1.12 (a) H0 : Septaphine is just as effective as Cephaline.H1 : Septaphine is more effective than Cephaline.

(b) (i) H1.(ii) Yes. A Type I error: to conclude that Septaphine is more

effective than Cepahline when, in fact, they are equallyeffective.

the language of statistical decision making

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