Hypothesis Testing

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Lecture 8

Hypothesis Testing

Statistics for

Civil & Environmental Engineers

Relax a little and think about issue today!

Making Decisions

Do materials meet specifications?

Have pollutant levels increased?

Has streamflow been affected by urbanization?

Does new blend result in greater strength concrete?

Does SO₂affect human health?

Is acid rain causing environmental damage?

Have new management procedures improved production?

How do we organize information?

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Procedure for Testing

1. Declare a null hypothesis which is the hypothesis to be tested

2. An alternative hypothesis which we really wish to test

3. Determine a test statistic

4. Determine a level of significance α based on the known distribution of the test statistics

5. Define a rejection (or critical) region

6. Use the observed data to verify whether the computed value of the test statistic is within or outside the rejection region

Statistics for Civil & Environmental Engineers

Type I error

- The null hypothesis is rejected when it should be accepted

Type II error

- The null hypothesis is not rejected when it is not true

Probabilities of Type I and Type II Errors(1)

Definition

Reality

Decision H₀true H₁true

H₀accepted H₀rejected

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Type I error depends on probability α

Type II error depends on α, the sample size, the true value of parameters

Probabilities of Type I and Type II Errors(2)

Properties

Null Hypothesis H₀: µ₁= µ₂

One-tailed Tests:

H₁: µ₁< µ₂(lower tail test) Rejection Region: X_test≤ -x_α

H₁: µ₁> µ₂(upper tail test) Rejection Region: X_test≥ -x_α

Type of Tests(1)

f(x|H0) α

-xα

f(x|H₀) α

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Type of Tests(2)

Two-tailed Test:

H₁: µ₁≠µ₂ Rejection Region: |X_test| ≥-x_α

Equivalently, if X_test≥ x_α/2or X_test≤ -x_α/2

f(x|H₀)

α/2 X_α/2 -xα/2

α/2

Null hypothesis H₀: µ=µ₀

Alternative hypothesis H₁: µ≠µ₀

Two-tailed Example (1)

Hypothesis

Test Statistic

Standardized Test Statistic

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Type II error for a given α

The probability βof a Type II error is dependent on α, n, and c/σ

1-α β

α 1-β

Accept H0

Accept H1

H₀true H1 true

For the same sample size n, β decrease as c/σ increases β decreases as the sample size nincreases

Characteristic Curves

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The complement of β

Probability of rejecting the null hypothesis when it is not true

Power Function

Properties

Power Function Example

Power Curve Example

Definition: Probability we do the right thing

1-α β

α 1-β

Accept H₀

Accept H₁

H0 true H₁true

One-tailed Example 1: Testing n=1 beam (1)

Do we have the premium beams, or the regular beams?

Testing n=1 beam

Premium: X ~ N[12, 1²] Regular: X ~ N[11, 1²]

State #1 – H₀: µ= 12 (σ=1) State #2 – H₁: µ= 11 (σ=1)

8 9 10 11 12 13 14 15

f(x|H₁) f(x|H0)

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Need a Decision Rule – Once a week test a beam:

Accept H₀: µ= 12 if X > c_X Accept H₁: µ= 11 if X ≤ c_X

c= critical x-value for test X ≤ c is rejection regionfor H₀

Type I error

One-tailed Example 1: Testing n=1 beam (2)

Type II error

β= P[Accept H₀| H₀false]

f(x|H₁) f(x|H0)

α c β

1-α β

α 1-β

Accept H0

Accept H1

H₀true H1 true

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c_X α β

9 0.0013 0.98

10 0.023 0.84

11 0.16 0.50

12 0.50 0.16

13 0.84 0.023

The Trade-off

Special values

10.72 0.10 0.76

9.67 0.01 0.91

β α

0.5

11 12

Now, what if we select n=4 beams to test?

Do we have the premium beams, or the regular beams?

Use sample average to construct a decision procedure

X ~ N → X ~ N

E[X]= E[X] ; Var[X]= σ²/n

Premium: X ~ N[12, (1/4)]

Regular: X ~ N[11, (1/4)]

One-tailed Example 2: Testing n=4 beam (1)

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New Trade-offs

c_X α β

10 3x10^-5 0.98

10.84 0.01 0.63

11 0.023 0.50

11.36 0.10 0.24

12 0.50 0.023

X₁, X₂~ normal σ₁and σ₂are known

Testing The Difference Between Two Means Using Known Variances

Assumptions

Hypothesis

Test Statistic

where

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X₁, X₂~ normal σ₁= σ₂

σ₁and σ₂are unknown

Testing The Difference Between Two Means When The Variances are Unknown But Equal

Assumptions

Hypothesis

Test Statistic

where

X₁~ N(µ₁, σ₁) and X₂~ N(µ₂, σ₂) σ₁≠ σ₂

σ₁and σ₂are unknown

Testing The Difference Between Two Means When The Variances are Unknown and Unequal

Assumptions

Hypothesis

Test Statistic

Degree of freedom:

Behrens-Fisher Problem

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Testing The Difference Between Two Means Example

For others, see Table 5.4.1

Main steps

- The ranking of a sample of data

- Division into a number of classes depending on the magnitudes and the range

- The fitting of a probability distribution Test Statistic

Goodness-of-Fit Tests(1)

Chi-Squared Goodness-of-Fit Test

where O_i: observed frequency, E_i: expected frequency

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Chi-Square Example(1)

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Main steps

- A completely specified theoretical continuous cdf:

- The empirical or sample distribution function:

- The test statistics:

- Reject if D> D_n,α(Table C.7)

Goodness-of-Fit Tests(2)

Kolmogorov-Smirnov Goodness-of-Fit Test

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Kolmogorov-Smirnov Example(1)

Kolmogorov-Smirnov Example(2)

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Test Statistics

Goodness-of-Fit Tests(3)

PPCC Test

where, x_i: ranked observation, w_i: fitted qunatile(=G^-1(1-q_i)) r: correlation coefficient, q_i(=p_i): plotting position G(x): proposed cdf for the events

Goodness-of-Fit Tests(4)

Lower Critical Values for PPCC Test

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