1. Chapter 9 Tests of Hypothesis
Single Sample Tests
The Middle Game – applications to the real world
Chapter 9B

3. Recall?
If Z1, Z2, ..., Zn are independent standard normal random variables, then
chi-square distribution
with n degrees freedom
t-distribution
with n degrees freedom
F-distribution
with n degrees freedom in
the numerator and m degrees
of freedom in the denominator

4. 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown
One-Sample t-Test

5. 9-3.1 Hypothesis Tests on the Mean
Figure 9-9 The reference distribution for H0:  = 0 with critical region for (a) H1:   0 , (b) H1:  > 0, and (c) H1:  < 0.

6. Tests on the mean - variance unknown
Peter Perk is concerned about the amount of caffeine in his pop (soft drink). It appears that his midday soda is keeping him awake during the boring statistics lectures. The label on the can states that it contains only 20 mg of caffeine. Peter Perk doesn’t believe this.
A random sample and testing of 25 cans of soda resulted in the following caffeine levels:
Test at a 2 percent level of significance.
H0:  = 20
H1:  > 20

7. The t-test One can of soda contains: About 10 teaspoons of sugar
150 calories
30-55 mg of caffeine
Artificial food colors and sulphites

8. The Prob-Value H0:  = 20 H1:  > 20
conclusion: Peter should be able to sleep
during the boring lectures.

9. Non-Central t Distribution – a Small Complication
If the alternative hypothesis is true, the statistic T0 does not have a mean of zero.
It has what is called the non-central t distribution.
Many of the operations performed for the central t must now be done using numerical techniques, e.g. integrating the non-central t distribution.
No sweat for us. The results we need are tabulated – we need the O.C. curves!
estimated
noncentrality
parameter

10. Non-Central t Distribution – a Small Complication
O.C. curves relate b, n, and d, where d=|m-m0|/s. and b is the Type II error probability.
Since variance is unknown, estimate s – either from previous experiments where we measured s, or after the current measurement set is collected.
Alternatively we can think of d as a certain number of standard deviations (if a relative measure is satisfactory).

11. O.C. Curves for t-test
two-sided t-test
with  = .05

12. More on Peter Perk’s Pop
If the actual caffeine level is 22 mg per can, what is the probability of not rejecting at the 5 percent level (i.e. the probability of a Type II error)?
d = |22-20|/5 = .4
n = 25
Pr{do not reject|  = 22}  .5

13. 9-4.1 Hypothesis Test on the Variance

14. 9-4.1 Hypothesis Test on the Variance

15. More on Variance Tests

16. Much more on Variance Tests

17. A Variance Test Example
Professor Vera Vance asserts that the variability of the IQ’s among the Engineering Management (ENM) students is significantly less than for the population at large.
It is well know that the distribution of IQ’s is normal with a mean of 100 and a standard deviation of 15.
Thirty ENM students were forced to take an exhaustive IQ test in which the sample standard deviation was computed to be
Professor Vance is willing to test her assertion at the 10 percent level.

18. Add a little variance to your life
H0: 2 = 152
H1: 2 < 152
Professor Vera Vance lecturing
the students on their excessive variability.

19. Prob-value and so much more…
Type II Error and Choice of Sample Size:
Operating characteristic curves are provided in Charts VIIi through VIIn

20. O.C. Curves for Variance

21. More sugar?
Problem 9-75
milligrams2

22. Problem 9-75 P-Value

23. Problem 9-75 Confidence Interval

24. Problem 9-75 Assume 2 = 40 what sample size to detect difference with prob = .9?
This is the O.C. graphic for alpha = .01

25. 9-5.1 Large-Sample Tests on a Proportion
Many engineering decision problems include hypothesis testing about p.
An appropriate test statistic is

26. 9-5 Tests on a Population Proportion
Another form of the test statistic Z0 is or
the sample proportion

27. Test on a Proportion - Example
A cable news commentator has asserted that more than half the population believe that the United States should not have taken military action against Iraq. Test this assertion at the 5 percent level.
CBS News Poll. Sept , N=706 adults nationwide. MoE ± 4 (for all adults).
"Looking back, do you think the United States did the right thing in taking military action against Iraq, or should the U.S. have stayed out?"
Right Thing
Stayed Out
Unsure %
9/14-16/07 39 53 8

28. An Example Continued
H0: p  .5
H1: p > .5
 = .05

29. 9-5.2 Type II Error and Choice of Sample Size
For a two-sided alternative where p is the true value
If the alternative is p < p0
If the alternative is p > p0

30. Type II Error
H0: p  .5
H1: p > .5
 = .05

31. 9-5.3 Type II Error and Choice of Sample Size
For a two-sided alternative
For a one-sided alternative

32. Sample Size Calculations
H0: p  .5
H1: p > .5
 = .01, =.05 when p = .55

33. Goodness-of-fit (GOF) Tests
Testing for the distribution of the underlying population

34. 9-7 The Chi-Square GOF Test
Assume there is a sample of size n from a population whose probability distribution is unknown.
Let Oi be the observed frequency in the ith class interval.
Let Ei be the expected frequency in the ith class interval.
The test statistic is chi-square with
df = k – p – 1 where p = number of estimated parameters

35. Example 9-12

36. More of Example 9-12

37. Much More of Example 9-12

38. Still Example 9-12

39. Yes, Example 9-12

40. The last of Example 9-12

41. How to do it with two samples!
Next Week – Chapter 10
How to do it with two samples!