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where B(c; n1, p0) is the cumulative binomial distribution To nd the critical value for a given , we would select the largest c satisfying B(c; n1, p0) The type II error calculation is straightforward Let p1 be an alternative value of p, with p1 p0 If p p1, X is Bin (n, p1) Therefore P 1Type II error when p p1 2 P 3X c when X is Bin 1n, p1 2 4 1 B 1c; n, p1 2 where B(c; n, p1) is the cumulative binomial distribution Test procedures for the other one-sided alternative H1: p p0 and the two-sided alternative H0: p p0 are constructed in a similar fashion For H1: p p0 the critical region has the form x c, where we would choose the smallest value of c satisfying 1 B(c 1, n, p0) .

For the two-sided case, the critical region consists of both large and small values Because c is an integer, it usually isn t possible to de ne the critical region to obtain exactly the desired value of To illustrate the procedure, let s reconsider the situation of Example 9-10, where we wish to test H0: p 005 verses H1: p 005 Suppose now that the sample size is n 100 and we wish to use 005 Now from the cumulative binomial distribution with n 50 and p 005, we nd that B(0; 100, 005) 00059, B(1; 100, 005) 00371, and B(2; 100, 005) 01183 (Minitab will generate these cumulative binomial probabilities) Since B(1; 100, 005) 00371 005 and B(2; 100, 005) 01183 005, we would select c 1.

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When the TraceSwitch is set to 4, all the levels will be output. If the switch is set to 3, levels 1 through 3 will output, and so on. The following code example uses the TraceSwitch:

Therefore the null hypothesis will be rejected if x 1 The exact signi cance level for this test is 00371 To calculate the power of the test, suppose that p1 003 Now 1 B 1c; n, p1 2 1 B 11; 100, 0032 1 01946 08054 and the power of the test is only 01946 This is a fairly small power because p1 is close to p0..

INTRODUCTION 10-1 10-2 INFERENCE FOR A DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES KNOWN 10-2.1 Hypothesis Tests for a Difference in Means, Variances Known 10-2.2 Choice of Sample Size 10-2.3 Identifying Cause and Effect 10-2.4 Con dence Interval on a Difference in Means, Variances Known 10-3 INFERENCE FOR A DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN 10-3.1 Hypothesis Tests for a Difference in Means, Variances Unknown 10-3.2 More about the Equal Variance Assumption (CD Only) 10-3.3 Choice of Sample Size 10-3.4 Con dence Interval on the Difference in Means

Imports DiagTrace = System.Diagnostics.Trace Public Class WebForm1 Inherits System.Web.UI.Page Shared myTrace As New TraceSwitch( _ myTraceSwitch , This is a Trace switch ) Private Sub Button1_Click( _ ByVal sender As System.Object, _ ByVal e As System.EventArgs) Handles Button1.Click Debug.WriteLineIf( _ myTrace.TraceVerbose, This is the Verbose Trace = 4 ) Debug.WriteLineIf( _ myTrace.TraceInfo, This is the Info Trace = 3 ) DiagTrace.WriteLineIf( _ myTrace.TraceWarning, This is the Warning Trace = 2 ) DiagTrace.WriteLineIf( _ myTrace.TraceError, This is the Error Trace = 1 ) End Sub End Class

10-4 PAIRED t-TEST 10-5 INFERENCES ON THE VARIANCES OF TWO NORMAL POPULATIONS 10-5.1 The F Distribution 10-5.2 Development of the F Distribution (CD Only) 10-5.3 Hypothesis Tests on the Ratio of Two Variances 10-5.4 -Error and Choice of Sample Size

5 A 6 7

10-5.5 Con dence Interval on the Ratio of Two Variances 10-6 INFERENCE ON TWO POPULATION PROPORTIONS 10-6.1 Large-Sample Test for H0: p1 10-6.2 Small-Sample Test for H0: p1 p2 (CD Only) 10-6.3 -Error and Choice of Sample Size p2 10-6.4 Con dence Interval for p1 p2

The output of the code when the trace setting is set to 4 is as follows:

After careful study of this chapter, you should be able to do the following: 1. Structure comparative experiments involving two samples as hypothesis tests 2. Test hypotheses and construct con dence intervals on the difference in means of two normal distributions

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