1. Chapter 20 Testing Hypothesis about proportions
Example:
Metal Manufacturer
Ingots
20% defective (cracks)
After Changes in the casting process:
400 ingots and only 17% defective
Is this a result of natural sampling variability or there is a reduction in the cracking rate?

2. Hypotheses
We begin by assuming that a hypothesis is true (as a jury trial).
Data consistent with the hypothesis:
Retain Hypothesis
Data inconsistent with the hypothesis:
We ask whether they are unlikely beyond reasonable doubt.
If the results seem consistent with what we would expect from natural sampling variability we will retain the hypothesis. But if the probability of seeing results like our data is really low, we reject the hypothesis.

3. Testing Hypotheses Null Hypothesis H0
Specifies a population model parameter of interest and proposes a value for this parameter
Usually:
No change from traditional value
No effect
No difference
In our example H0:p=0.20
How likely is it to get 0.17 from sample variation?

4. Testing Hypotheses (cont.)
Normal Sampling distribution
How likely is to observe a value at least 1.5 standard deviations below the mean of a normal model
Management must decide whether an event that would happen 6.7% of the time by chance is strong enough evidence to conclude that the true cracking proportion has decreased

5. A Trial as a Hypothesis Test
The jury’s null hypothesis is
H0 : innocent
If the evidence is too unlikely given this assumption, the jury rejects the null hypothesis and finds the defendant guilty. But if there is insufficient evidence to convict the defendant, the jury does not decide that H0 is true and declare him innocent. Juries can only fail to reject the null hypothesis and declare the defendant “not guilty”

6. The Reasoning of Hypothesis Testing
To perform a hypothesis test, we must specify an alternative hypotheses. Remember we can never prove a null hypothesis, only reject it or retain it. If we reject it, we then accept the alternative
Example: Pepsi or Coke
p : proportion preferring coke
H0 : p = 0.50
HA : p ≠ 0.50

7. The Reasoning of Hypothesis Testing (cont.)
Plan
Specify the model and test you will use (proportions, means).
We call this test about the value of a proportion a one-proportion z-test
Mechanics
Actual Calculation of a test from the data.
P-value : the probability that the observed statistic value could occur if the null model were correct. If the P-value is small enough, we reject the null hypothesis

8. The Reasoning of Hypothesis Testing (cont.)
Conclusion
The conclusion in a hypothesis test is always a statement about the null hypothesis. The conclusion must state either that we reject or that we fail to reject the null hypothesis

9. Alternatives Two-sided Alternative One-sided Alternative
HA : p ≠ (Pepsi – Coke)
The P-value is the probability of deviating in either direction from the null hypothesis
One-sided Alternative
H0 : p = 0
HA : p < (Ingots)
The P-value is the probability of deviating only in the direction of the alternative away from the null hypothesis value.

10. Exercises
Page 467 #1 #3
#20

11. Chapter 21 More About Tests
Example : Therapeutic Touch (TT)
One-proportion z-test
15 TT practitioners 10 trials each
H0 : p=0.50
HA : p>0.50
Random Sampling
Independence
10% condition
Success/Failure condition
Observed proportion 0.467
Find the P-value…

12. How to Think About P-values
A P-value is a conditional probability. It is the probability of the observed statistic given that the null hypothesis is true.
P-value : P(Observed statistic value|H0)

13. Alpha Levels
When the P-value is small, it tells us that our data are rare given the null hypothesis.
We can define a “rare event” arbitrarily by setting a threshold for our P-value.
If our P-value falls below that point we’ll reject the null hypothesis. We call such results “statistically significant” the threshold is called an alpha level or significance level.

14. Alpha Levels (cont.) Rejection Region One Sided Two sided  = 0.10
 = 0.05
 = 0.01
Rejection Region
One Sided Two sided

15. Making Errors Type I error Type II error
The null hypothesis is true, but we mistakenly reject it.
Type II error
The null hypothesis is false but we fail to reject it.

16. Types of errors Examples Medical disease testing Jury Trial
I : False Positive
II : False Negative
Jury Trial
I : Convicting an innocent
II : Absolving someone guilty

17. Probabilities of errors
To reject H0, the P-value must fail below . When H0 is true that happens exactly with probability  so when you choose the level , you are setting the probability of a Type I error to .
When H0 is false and we fail to reject it, we have made a Type II error. We assign the letter  to the probability of this mistake

18. Power
The power of a test is the probability that it correctly rejects a false null hypothesis. When the power is high, we can be confident that we’ve looked hard enough.
We know that  is the probability that a test fails to reject a false null hypothesis, so the power of the test is the complement 1 - 
When we calculate power, we have to imagine that the null hypothesis is false. The value of the power depends on how far the truth lies from the null hypothesis value. We call this distance between the null hypothesis value p0 and the truth p the effect size.

19. Chapter 22 Comparing Two Proportions
Recall (Ch.16)
The variance of the sum or difference of two independent random quantities is the sum of their individual variances
Example of the cereals

20. Comparing Two Proportions (cont.)
The Standard Deviation of the Difference Between Two Proportions
For proportions from the data

21. Assumptions and Conditions
Random Sampling
10% condition
Independent Samples Condition
The two groups we are comparing must also be independent of each other (usually evident from the way the data is collected).
Example :
Same group of people before and after a treatment are not independent
Success and failure condition in each sample

22. The Sampling Distribution
The sampling distribution for a difference between two independent proportions
Provided the assumptions and conditions the sampling distribution of is modeled by a normal model with mean
and standard deviation

23. A two-proportion z-interval
When the conditions are met, we are ready to find the confidence interval for the difference of two proportions p1-p2. Using the standard error of the difference
The interval is
The critical value z* depends on the particular confidence level.

24. Exercises
Two-proportion z-interval
(page 493, 496)

25. Example Snoring Random sample of 1010 Adults From 995 respondents:
37% snored at least few nights a week
Splitting in two age categories:
Under 30 Over 30
26.1% of % of 811
Is the difference of 13.1% real or due only to sampling variability?

26. Example (cont. snoring)
H0 : p1 – p2 = 0
But p1 and p2 are linked from H0
p1 = p2
Pooling:
Combining the counts to get an overall proportion

27. Two-Proportion z-test
The conditions for the two-proportion z-test are the same as for the two-proportion z-interval . We are testing the hypothesis:
H0 : p1 = p2
Because we hypothesize that the proportions are equal, we pool them to find
And we use the pooled value to estimate the standard error

28. Two-Proportion z-test (cont.)
Now we find the test statistic using the statistic
When the conditions are met and the null hypothesis is true, this statistic follows the standard Normal model, so we can use that model to obtain a P-value

29. Example (cont. snoring)
Randomization 10% Condition
Independent samples condition Success / Failure
The P-value is the probability of observing a difference greater or equal to 0.131
The two sided P-value is This is rare enough, so we reject the null hypothesis and conclude that there us a difference in the snoring rate between this two age groups.

30. Exercise
Page 508 #16

31. Homework #5 Page 423 #8, 16 Page 443 #12, 18 Page 467 #2, 4, 6, 12