m34- hypothesis testing 1 department of ism, university of alabama, 1992-2003 homework chp 11. use...

M34- Hypothesis Testing 1 Department of ISM, University of Alabama, 1992-2003

Homework

Chp 11. Use equations on formula sheet. Page 417 # 5, 6, 8. (Use CI to make a decision).

Page 441 # 47, 49, 50. (Use CI to make . . . .)

Chp 15, Read 633 bottom, 634 middle.

Page 636 # 50 (Do part c to answer part b.)

# 51. (Do part c to answer part b.)

# 53, 54 (Review)

# 47, 48. (Give answer for p-value

(see answer to 47 ibob) in relation to the -level.)


Hypothesis Testing


Lesson Objectives

Understand the “types of errors” in decision making.

Know what the -level means.

Learn how to use “p-values” andconfidence intervals for decisionmaking.


Court case

Hypothesis:Hypothesis: Defendant is Defendant is innocentinnocent..Alternative:Alternative: Defendant is Defendant is guiltyguilty..

Decisions: Based on the sample data.

Reject Innocence

Reject Innocence

Declare“Guilty”“Guilty”Declare“Guilty”“Guilty”

PersonPersongoes togoes to

jail!jail!

PersonPersongoes togoes to

jail!jail!

Do not Reject Innocence

Do not Reject Innocence

Declare“Not “Not

Guilty” Guilty”

Declare“Not “Not

Guilty” Guilty”

PersonPerson goes goes

free!free!

PersonPerson goes goes

free!free!


Type I: Sending an innocent person to jail.

Type I: Sending an innocent person to jail.

Type II: Letting a guilty person go free.Type II: Letting a guilty person go free.

= level of risk deemed reasonablefor the occurrence of a Type I error.

= the point of “reasonable doubt.”

Types of Errors in a court case


Types of Errors, in general

Type I: Concluding that the hypothesized parameter value is wrong, but in reality it is correct.

Type I: Concluding that the hypothesized parameter value is wrong, but in reality it is correct.

Type II: Not concluding that the hypothesized parameter value is wrong, but in reality it is incorrect.

Type II: Not concluding that the hypothesized parameter value is wrong, but in reality it is incorrect.

= level of risk, chosen by the user,for allowing a Type I error to occur.

= risk for making Type II error.

Net weight of potato chip bagsshould be 16.00 oz.

An FDA inspector will take a random sample of 36 bags. If the net weight is too low, the chip company will be fined substantially.

From the FDA perspective, what would the Type I and Type IIerrors be (in words)?

Potato Chip Inspection by FDA


Type I: Penalizing the potato chip company when in reality they were NOT cheating the consumer.

Type I: Penalizing the potato chip company when in reality they were NOT cheating the consumer.

Type II: Not detecting that the potato chip company was cheating the consumers, when in reality theywere.

Type II: Not detecting that the potato chip company was cheating the consumers, when in reality theywere.

Potato Chips; types of errors

Which is more serious, from the FDA’s perspective?


Chose an -level that considersthe consequences of the Type I and Type II errors.

and are inversely related;as one goes up, the other goes down, but NOT by equal amounts.

Selecting an -level


Reject the hypothesized value if:

1. it is outside the confidence interval.

2. the p-value is less than the user specified -level. (p-value < -level)

3. the calculated test statistic valueis in the “critical region.”

Statistical Inference Methods:

Three methods; each should give the same result.Three methods; each should give the same result.


Decisions are based on

the datathe data.Wrong decisions are the result

of chance,

not mistakes.

(1- )100% Confidence Interval Method

Two tailed test;“Is the mean something other than 40.0?”

One tail test;“Is the mean something greater than 40.0?” or “Is the mean something less than 40.0?”

Hypothesized mean: 40.0

Result:Result: EachEach tail has the tail has the full full -level. -level. Use only ONEUse only ONE tailtail for making a decision. for making a decision.

.05.05 .95.95

.10.10 .90.90

.01.01 .99.99

.05.05 .90.90

.10.10 .80.80

.01.01 .98.98

DesiredDesired -level:-level:

Size ofSize ofCI to use: CI to use:

1 -

1 - 2

Result:Result: Each tail has half of Each tail has half of . .


p-Value Method

The probability of observing a future statistic value that is as big or more extreme, in the direction(s) of interest, than the value we just observed,assuming that the hypothesized value is the correct parameter.

Calculate p-value using the most appropriate distribution.Calculate p-value using the most appropriate distribution.

Decision rule:Decision rule: If p-value < If p-value < -level,-level, reject the hypothesized value. reject the hypothesized value.


X = 42.6

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

40 X42.6

Hypo. mean: 40.0,p-Value:

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

40 X

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

40 X

37.4

42.6

42.6

Two tailed test;“Is the mean something other than 40.0?”

Upper tail test;“Is the mean something greater than 40.0?”

Lower tail test;“Is the mean something less than 40.0?”

p-value / 2

p-value

p-value

p-value / 2

2.62.6


Sample results:X = 43.0s = 7.2

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

0 Z1.50

.4332

Z = 43.0 – 40.0 2.0

= 1.50

.5000

.0668

40 X43.0

X X

~ N( ?, 8.0)16

~ N( ?, 2.0)

XnX

Hypothesized mean: 40.0.Adjust machine if it’s offin either direction.

More extreme Also

more extreme

-1.50

p-value = .0668 •2 = .1336

than 3.0 unitsthan 3.0 units

Problem 1, using p-Value

What distributionshould be used?

Pick = .05

(p-value =.1336) > ( =.05); do not reject 40.0.



X X

~ N( ?, 8.0)16

~ N( ?, 2.0)

XnX

Hypothesized mean: 40.0.Adjust machine if it’s offin either direction.

Problem 1 with Confidence Interval


Pick = .05

40.0 falls inside the C.I.; do not reject 40.0.

X ± Z/2 n

43.0 ± 1.96 2.043.0 ± 4.92

(38.08, 47.92)



-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

0 t

t = 36.4 – 40.0 7.2 / 4

= -2.00

40 X36.4

X X

~ N( ?, ?)16

~ N( ?, ?)

XnX

Hypothesized mean: 40.0.FDA will fine companyif the mean is lower.

More extreme

-2.00

p-value = ..

than 3.6 units below.

Problem 2, using p-Value

From the t-table . . . .

From Excel,

.0320.0320

more than .025,more than .025,less than .050.less than .050.


(p-value =.0320) < (-level = .05); reject 40.0.

Pick = .05



X X

~ N( ?, ?)16

~ N( ?, ?)

XnX

Hypothesized mean: 40.0.FDA will fine companyif the mean is lower.


40.0 falls outside the C.I.; reject 40.0.

Pick = .05

X ± t, 15 sn

(33.245, 39.555)

36.4 ± 3.1554

36.4 ± 1.753 7.2

16

Problem 2 with Confidence Interval


Two people in different rooms. “A” is shown one of five cards, selected

randomly. “A” transmits his thoughts. “B” selects the card he thinks is being

sent to him, and records it The process is repeated 20 times;

the cards are shuffled each time.

Does a person have ESP?

Experiment:

X = a count of the number correct.X ~Bino(n=20, =.20)


X = a count of the number correct.X ~Bino(n=20, =.20) = n = 4.0 (Cannot use Normal approx.)

Data: “B” got 9 out 20 correct.Does “B” do better than guessing?

Hypothesized value: = .20,

p-value = P(X >= 9)= 1 – P(X <= 8)= 1 – .9900

= .0100 0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20X

P(X

= x

)Use Table A.2

-level =

Bino(20, 0.20)Use Binomial Dist.

.05.05

Reject .20; she does better!


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m34- hypothesis testing 1 department of ism, university of alabama, 1992-2003 homework chp 11. use...

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