How to calculate beam with slope deflection. Discuss the application of slope-deflection method to beams and frames. Analyse statically indeterminate structures using slope-deflection method
This is a lecture note on CFD subject taught at University of Melbourne. It focused on ODE topics. This is the second lecture note on this subject.Full description
Full description
Lecture 2 - Agency Theory
taxFull description
Detailed introduction and procedure of Transition curves
Full description
Atty Lim
Size Reduction Rev
Full description
rab
Deskripsi lengkap
IE 28 Statistical Analysis for Industrial Engineers
11.11.2010
1.
Review of Type I and Type II errors er rors
2.
Use of p-value (versus conventional test of hypothesis)
3.
Exercises
1.
Review of Type I and Type II errors er rors
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 – Simple – completely completely specifies the distribution Composite – Composite – does 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
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 1.
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
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
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 Correct 50g Weight of the packs is 50g Weight of the packs is Type I error not 50g Weight of the packs is Type II error Weight of the packs is 50g not 50g Weight of the packs is Correct not 50g
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
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 1.
Decision using P-values
If p-value > α
Do not reject H0
If p-value ≤ α
“If p is low, make it go”
Reject H0
Watch out for marginal cases
Note: Most statistical software refers to p-values
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
Test
α= 5% H0 : μ= 50 critical value: Zα/2=
Sample size = 16 Sample Standard deviation = 4 Sample Mean = 51.9
What is the p-value?
In Perspective
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
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?
