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

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

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

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

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

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

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.

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

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) = 0.0001 Compute Z = normsinv(.9999) = 3.71947 Reject the null hypothesis if: Test statistic > 3.71947

Heart Valve Example: conclusion Test statistic = 5.404 Reject null hypothesis if test statistic > 3.71947 We should REJECT the null hypothesis

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

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

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

Chi-Test Examples See 95 Murders.xls spreadsheet