Hypothesis Testing

• Approach 1 - Fixed probability of Type I 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. Compute 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. Draw an engineering or scientific

conclusion.

Hypothesis Testing

• Approach 2 - Significance testing (P-value approach)

1. State the null and alternative hypotheses.

2. Choose an appropriate test statistic.

3. Compute value of test statistic and determine P-value.

4. Draw conclusion based on P-value.

P = 0 P = 10.25 0.5 0.75

Single Sample Test of the Mean

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.

H0: ________ n = ________

H1: ________ σ unknown

use ___ distribution.c

Example (cont.)Approach 2:

= _________________

Using Excel’s tdist function,

P(x ≤ -1.118) = _____________

Conclusion: __________________________________

nS

XT

/

Example (concl.)

Approach 1:

t0.05,19 = _____________

Since H1 specifies “< μ,” tcrit = ___________

tcalc = _________

Conclusion: _________________________________

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.

• In other words:– Failure to reject H0 does not imply that μ is equal to the

stated value, only that we do not have sufficient evidence to favor H1.

Your turn …

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 at an α level of 0.05. Draw the picture.

Homework for Wednesday, Nov. 3

• pp. 298-299: 5, 12, 15

• pp. 319-323: 3, 4, 6, 7, 8