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M 562 Section-10-4

# M 562 Section-10-4 - MTH/STA 562 CALCULATING TYPE II ERROR...

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Unformatted text preview: MTH/STA 562 CALCULATING TYPE II ERROR PROBABILITIES AND FINDING THE SAMPLE SIZE FOR THE Z TEST Consider testing the null hypothesis H : & = & against the alternative hypothesis H a : & = & a > & where & and & a are speci&ed values of & , with the rejection region RR = n Y > k o for some choice of k . Then, by de&nition, ¡ = P f Type I Error g = P f rejecting H when H is true g = P f Value of test statistic is in RR when H is true g = P n Y > k when & = & o = P ( Y & & ¢= p n > k & & ¢= p n ) ¡ P ( Z > k & & ¢= p n ) Thus, k & & ¢= p n = z & or k = & + z & ¢ p n (10 : 4 : 1) On the other hand, by de&nition, £ = P f Type II Error g = P f accepting H when H is false g = P f Value of test statistic is not in RR when H is false g = P n Y ¢ k when & = & a o = P ( Y & & a ¢= p n ¢ k & & a ¢= p n ) ¡ P ( Z ¢ k & & a ¢= p n ) Thus, k & & a ¢= p n = & z ¡ or k = & a & z ¡ ¢ p n (10 : 4 : 2) From (10 : 4 : 1) and (10 : 4 : 2), we conclude that & + z & ¢ p n = & a & z ¡ ¢ p n or,equivalently,...
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M 562 Section-10-4 - MTH/STA 562 CALCULATING TYPE II ERROR...

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