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Lecture on Type 2 errors and determining samples size
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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