1. Sample Size Needed to Achieve High Confidence (Means)
Considering estimating X, how many observations n are needed to obtain a 95% confidence interval for a particular error tolerance?
The error tolerance E is ½ the width of the confidence interval
Here,  is a conservative (high) estimate of the true std dev X, often gotten by doing a preliminary small sample
1.960 can be adjusted to get different confidences

2. Derivation of Formula E = z * std. error = z * sigma/sqrt(n)
Thus, sqrt(n) = z * sigma/E
So, n = [z * sigma/E] 2

3. Sample Size Needed to Achieve High Confidence (Proportions)
Considering estimating pX, how many observations n are needed to obtain a 95% confidence interval for a particular error tolerance?
The error tolerance E is ½ the width of the confidence interval
Here, p is a conservative (closer to 0.5) estimate of the true population proportion pX, often gotten by doing a preliminary small sample
1.960 can be adjusted to get different confidences

4. Derivation E = z * std. error But now, std error = sqrt[p(1-p)/n], so
E = z * sqrt[p(1-p)]/sqrt(n), and hence
Sqrt(n) = z * sqrt[p(1-p)]/E,
=> n = [z *sqrt[p(1-p)]/E]2

8. Hypothesis Testing
Formulate null hypothesis (action is associated with alternative)
Compute Test Statistic
Determine Acceptance Region

9. Heart Valve Example Original yield = 52%
Change Process (Sort in batches of 5)
Sample 100 assemblies: sample yield = 79%

10. Heart Valve Example: Formulate Null Hypothesis
Null Hypothesis: the true process yield is 52% or less Why?????
Note: Action would be associated with the alternative----adopt new process only if it really increases the yield.

11. Heart Valve Example: Compute Test Statistic
We will use the Normal Approximation:
(how many standard deviations to the right of .52 is .79?)

12. Heart Valve Example: Determine Acceptance Region
Suppose we want to set the probability of rejecting the null hypothesis given that it is true (type 1 error) =
Compute Z = normsinv(.9999) =
Reject the null hypothesis if:
Test statistic >

14. Heart Valve Example: conclusion
Test statistic = 5.404
Reject null hypothesis if test statistic >
We should REJECT the null hypothesis

27. Two-Sample Tests for Means
Used when we wish to compare the means of two populations when both means are unknown
Tools => Data Analysis => two sample test
See Two Sample Spreadsheet

28. Two Sample Tests
Z-test: two sample for means – when std. deviations are known for both distributions
T-test: two sample assuming equal variances
T-test: two sample assuming unequal variances
Usually just use unequal variance test

39.   Important Note: The use of the chi-square test on variance requires that the underlying population be normally distributed.

53. Chi-Test Examples
See 95 Murders.xls spreadsheet