JMB Ch10 Lecture 1 9th ed. v Mar 2012 EGR 252 Spring 2012 Slide 1

Statistical Hypothesis Testing

A statistical hypothesis is an assertion concerning one or more populations.

In statistics, a hypothesis test is conducted on a set of two mutually exclusive statements:

H0 : null hypothesis

H1 : alternate hypothesis Example

H0 : μ = 17

H1 : μ ≠ 17 We sometimes refer to the null hypothesis as the

“equals” hypothesis.

JMB Ch10 Lecture 1 9th ed. v Mar 2012 EGR 252 Spring 2012 Slide 2

Tests of Hypotheses - Graphics I We can make a decision about our hypotheses

based on our understanding of probability. We can visualize this probability by defining a

rejection region on the probability curve. The general location of the rejection region is

determined by the alternate hypothesis.

H0 : μ = _____ H1 : μ < _____ H0 : μ = _____

H1 : μ ≠ _____

H0 : p = _____ H1 : p > _____

One-sided Two-sided

One-sided

JMB Ch10 Lecture 1 9th ed. v Mar 2012 EGR 252 Spring 2012 Slide 3

Choosing the Hypotheses

Your turn …Suppose a coffee vending machine claims it dispenses an 8-oz cup of coffee. You have been using the machine for 6 months, but recently it seems the cup isn’t as full as it used to be. You plan to conduct a statistical hypothesis test. What are your hypotheses?

H0 : μ = _____ H1 : μ ≠ _____

H0 : μ = _____H1 : μ < _____

JMB Ch10 Lecture 1 9th ed. v Mar 2012 EGR 252 Spring 2012 Slide 4

Potential errors in decision-making

α Probability of committing a

Type I error Probability of rejecting the

null hypothesis given that the null hypothesis is true

P (reject H0 | H0 is true)

β Probability of committing a

Type II error Power of the test = 1 - β

(probability of rejecting the null hypothesis given that the alternate is true.)

Power = P (reject H0 | H1 is true)

H0 True

H0 False

Do not reject H0

Correct

Decision

Type II error

Reject H0

Type I error

Correct

Decision

JMB Ch10 Lecture 1 9th ed. v Mar 2012 EGR 252 Spring 2012 Slide 5

Hypothesis Testing – Approach 1

Approach 1 - Fixed probability of Type 1 error.

1. State the null and alternative hypotheses.

2. Choose a fixed significance level α.

3. Specify the appropriate test statistic and establish the critical region based on α. Draw a graphic representation.

4. Calculate the value of the test statistic based on the sample data.

5. Make a decision to reject or fail to reject H0, based on the location of the test statistic.

6. Make an engineering or scientific conclusion.

JMB Ch10 Lecture 1 9th ed. v Mar 2012 EGR 252 Spring 2012 Slide 6

Hypothesis Testing – Approach 2Significance testing based on the calculated P-value

1. State the null and alternative hypotheses.2. Choose an appropriate test statistic.3. Calculate value of test statistic and determine P-

value. Draw a graphic representation.

4. Make a decision to reject or fail to reject H0, based on the P-value.

5. Make an engineering or scientific conclusion.

Three potential test results:

P-value 0 1.000.25 0.50 0.75

p = 0.02 ↓P-value

p = 0.45 ↓ p = 0.85 ↓

JMB Ch10 Lecture 1 9th ed. v Mar 2012 EGR 252 Spring 2012 Slide 7

Hypothesis Testing Tells Us …Strong conclusion:

If our calculated t-value is “outside” tα,ν (approach 1) or we have a small p-value (approach 2), then we reject H0: μ = μ0 in favor of the alternate hypothesis.

Weak conclusion: If our calculated t-value is “inside” tα,ν (approach 1)

or we have a “large” p-value (approach 2), then we cannot reject H0: μ = μ0.

Failure to reject H0 does not imply that μ is equal to the stated value (μ0), only that we do not have sufficient evidence to support H1.

JMB Ch10 Lecture 1 9th ed. v Mar 2012 EGR 252 Spring 2012 Slide 8

Example: Single Sample Test of the Mean P-value Approach

A sample of 20 cars driven under varying highway conditions achieved fuel efficiencies as follows:

Sample mean x = 34.271 mpg

Sample std dev s = 2.915 mpg

Test the hypothesis that the population mean equals 35.0 mpg vs. μ < 35.

Step 1: State the hypotheses.H0: μ = 35

H1: μ < 35

Step 2: Determine the appropriate test statistic.

σ unknown, n = 20 Therefore, use t distribution

JMB Ch10 Lecture 1 9th ed. v Mar 2012 EGR 252 Spring 2012 Slide 9

Single Sample Example (cont.)Approach 2:

= -1.11842

Find probability from chart or use Excel’s tdist function.

P(x ≤ -1.118) = TDIST (1.118, 19, 1) = 0.139665

p = 0.14

0______________1 Decision: Fail to reject null hypothesis

Conclusion: The mean is not significantly less than 35 mpg.

nS

XT

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JMB Ch10 Lecture 1 9th ed. v Mar 2012 EGR 252 Spring 2012 Slide 10

Example (concl.)

Approach 1: Predetermined significance level (alpha)

Step 1: Use same hypotheses.

Step 2: Let’s set alpha at 0.05.

Step 3: Determine the critical value of t that separates the reject H0 region from the do not reject H0 region.

t, n-1 = t0.05,19 = 1.729

Since H1 format is “μ< μ0,” tcrit = -1.729

Step 4: tcalc = -1.11842

Step 5: Decision Fail to reject H0

Step 6: Conclusion: The population mean is not significantly less than 35 mpg.

******Do not conclude that the population mean is 35 mpg.******