11.11.2010

IE 28

Statistical Analysis for Industrial Engineers

Agenda

1. Review of Type I and Type II errors

2. Use of p-value (versus conventional test of hypothesis)

3. Exercises

Statistical Hypothesis

Definition

Statistical definition

an assertion about the distribution of one or more random

variables

an assertion about the parameters of a distribution or a model.

It is a statement that needs to be proven or disproven

Two types of statistical hypotheses:

Simple – completely specifies the distribution

Composite – does not completely specify the distribution

Hypothesis Testing

A test of statistical hypothesis is a rule which when the

experimental values have been obtained, leads to a decision

to reject or not reject the hypothesis under consideration

The critical region, C, is that subset of the sample space

which leads to the rejection of the hypothesis under

consideration

The construction and choice of this critical region are what

make up the test of hypothesis.

Steps on Hypothesis Testing

1. State null and alternative hypothesis

2. Choose and compute for the test

statistic

3. Determine critical area/acceptance

region

4. Compare test statistic and critical

region to make conclusion

Example Suppose that X, is a random variable, an outcome of a random

experiment

We want to test if a pack really weighs 50g as said in the wrapper

Example

The random experiment is the M&M pack with X denoting

its weight.

We assume that X is normally distributed

μ0

σ = 2.5

In our test, we would accept it as 50 grams when it goes in

this interval

(48.5, 51.5)

Example

Null Hypothesis Alternative Hypothesis

The weight of an M&M

pack is 50 grams

μ = 50

The weight of an M&M

pack is not 50 grams

μ ≠ 50

Type I and Type II Errors

Definition

Type I Error Type II Error

Rejection of the Null

Hypothesis when it is

true

Alpha(α)

Failure to reject the

null hypothesis when it

is false

Beta(β)

Power of the Test (1- β)

Probability of rejecting

H0 when it is false

Properties of Type I and Type II Error

A decrease in the probability of on error generally results an increase in the probability of the other

The size of the critical region, and therefore the probability of committing a type I error, can always be reduced by adjusting the critical values

Summary of Type I and Type II Errors

Possible situations in Testing a Statistical

Hypothesis

M&M’s Example

Situation Conclusion from

the Experiment

Type of

Conclusion

Weight of the packs is

50g

Weight of the packs is

50g Correct

Weight of the packs is

not 50g Type I error

Weight of the packs is

not 50g

Weight of the packs is

50g Type II error

Weight of the packs is

not 50g Correct

Computation for Type I Error

M&M’s Example

Suppose that we are getting 10 M&M’s packs to test if our

hypothesis is correct or not.

What is the Type I Error?

What if…

We widen the

acceptance region

to (48, 52)

What will be the

Type I error?

We increase our

sample size to 16

What will be the

Type I Error?

Insights on Type I Error

We could reduce the Type I Error value by

Widening the acceptance region

Increasing the sample size

Computation for Type II Error

M&M’s Example We use the new acceptance region (48,52)

What if the weight of the pack is really 52g and not 50g

The variance is still the same.

Sample Size is still 10

What if…

What if the weight of the pack is really 50.5g and not

50g

The variance is still the same.

Sample size is 16

Take Note

1 - β

Power of the Test

Probability of rejecting the null hypothesis

H0 when the alternative hypothesis is true

Measure of the sensitivity of a statistical

test

Summary of all the parameters for the

M&M’s Example

Additional Concepts

Type I Error is related to the “rejection” region

(area on the fringes)

Type II Error is related to the “acceptance”

region (area inside)

It would be impossible to compute Type II error

without a specific alternative (being true)

P-values

Definition

Smallest level of significance that would lead to the

rejection of the null hypothesis with the given data

Lowest level of significance at which the observed

value of the test statistic (TS) is significant

Present convention require the pre-selection of the

level of significance α (5%, 1%) and choosing the

critical region accordingly

Steps in Test of Hypothesis: P-values

1. State null and alternative hypothesis

2. Choose and compute for the test

statistic

3. Compute p-value based on the test

statistic

4. Use judgement to conclude based on

the p-value

Decision using P-values

If p-value > α Do not reject H0

If p-value ≤ α Reject H0

Watch out for marginal cases

Note: Most statistical software refers to p-values

“If p is low,

make it go”

More on p-values

P-values are actually difficult to

compute except for the standard

normal distribution (Z)

P-values inform us how well the TS falls

into the critical region

Using the P-values preclude the need to

determine a level of significance

Example Consider the case of a two tailed test with

α= 5%

H0 : μ= 50

critical value: Zα/2=

Test

Sample size = 16

Sample Standard deviation = 4

Sample Mean = 51.9

What is the p-value?

In Perspective

Critical Region

Exercises

Problem 1

Suppose and allergist wishes to test the

hypothesis that at least 30% of the

public is allergic to some cheese

products. Explain how the allergist

could commit

Type I Error

Type II Error

Problem 2

A sociologist is concerned about the effectiveness

of a training course designed to get more drivers

to use seatbelts in automobiles.

What hypothesis is she testing is she commits a

type I error by erroneously concluding that the

training course is ineffective?

What type of hypothesis is she testing if she

commits a type II error by erroneously

concluding that the training course is effective?

Problem 3 The proportion of adults living in a small town who are

college graduates is estimated to be p=0.6. to test this hypothesis, a random sample of 15 is selected. If the number of college graduates in our sample is anywhere from 6 to 12, we will fail to reject the null hypothesis that p=0.6; otherwise we shall conclude that p is not equal to 0.6.

Evaluate α assuming p=0.6, using the binomial distribution.

Evaluate β for the alternatives p=0.5 and p=0.7. what about if p=0.59. What does this show?

Problem 4

Repeat the previous exercise when 200

adults are selected and the acceptance

region is defined to be 110<x<130

where x is the number of college

graduates in our sample. Use the

normal to binomial distribution.

Problem 5

A random sample of 400 samples in a certain city asked

if they favor an additional 4 gasoline sales tax to provide

badly needed revenues for street repairs. If more than

220 but fewer than 260 favor the tax, we shall conclude

that 60% of the voters are for it.

Find the probability of committing a type I error if

60% of the voters favor the increased tax.

What is the probability of committing a type II error

using this test if actually only 48% of the voters are in

favor of the additional gasoline tax.

Problem 6

A consumer products company is formulating a new

shampoo and is interested in foam height (in millilitres).

Foam height is approximately normally distributed and

has a standard deviation of 20 millilitres. The company

wishes to test H0: μ= 175 mL versus H1: μ> 175mL,

using the results of 10 samples.

Find the type I error probability, if the critical region is

x > 185mL

What is the probability of Type II error if the true

mean foam height is 195mL?

HOMEWORK

Solve the following problems

Montgomery

9-6

9-8

9-15

9-19

fin