MTB > # Solution to Exercise 23       (STAT 657, Fall 2008, Fall 2010, & Fall 2012)
 MTB >  
 MTB > # I'll put the size of the sign test in k1, and the power of the sign test
 MTB > # against the alternative -0.1 in k2.
 MTB > cdf 43 k1;
 SUBC> bino 100 0.5.
 MTB > name k1 'signsize' k2 'signpow'
 MTB > print k1
 
 signsize 0.0966740

 MTB > # For the case of mu = -0.1, I'll put P( X_i > 0 ) in k3.
 MTB > name k3 'prob'
 MTB > cdf 0 k3;
 SUBC> norm -0.1 1.
 MTB > let k3 = 1 - k3
 MTB > print k3
 
 prob     0.460172

 MTB > cdf 43 k2;
 SUBC> bino 100 k3.
 MTB > print k2
 
 signpow  0.307577

 MTB > # Letting alpha be the size of the sign test, I'll put -z_alpha in k4.
 MTB > invcdf k1 k4;
 SUBC> norm 0 1.
 MTB > name k4 '-z_alpha'
 MTB > print k4
 
 -z_alpha -1.30074

 MTB > # It follows from near the top of p. 5-10 of the class notes that
 MTB > #       k2 = Phi( k4 -sqrt(n)*eta ) = Phi( k4 + sqrt(n)/10 ),
 MTB > # from which it follows that
 MTB > #   Phi^{-1}(k2) = k4 + sqrt(n)/10,
 MTB > # or equivalently,
 MTB > #   sqrt(n) = 10[ Phi^{-1}(k2) - k4 ].

 MTB > # I'll solve the above for n and put the value in k5.
 MTB > invcdf k2 k5;
 SUBC> norm 0 1.
 MTB > let k5 = 10*( k5 - k4 )
 MTB > let k5 = k5*k5
 MTB > name k5 'n'
 MTB > print k5
 
 n        63.6818

 MTB > # This is very close to the value 63.6620 which results from using the sample
 MTB > # size suggested by the A.R.E. of the sign test w.r.t. the z test when the
 MTB > # underlying distribution is norml.  (The A.R.E. is 2/pi, or about 0.636620.)

 MTB > # As a check that 64 is the best choice for a sample size, I'll put the power
 MTB > # of the z test for various sample sizes (63, 64, and 65) in k6 through k8,
 MTB > # and compare these power values against the power of the specified sign test.
 MTB > let k16 = k4 + sqrt(63)/10
 MTB > let k17 = k4 + sqrt(64)/10
 MTB > let k18 = k4 + sqrt(65)/10
 MTB > cdf k16 k6;
 SUBC> norm 0 1.
 MTB > cdf k17 k7;
 SUBC> norm 0 1.
 MTB > cdf k18 k8;
 SUBC> norm 0 1.
 MTB > name k6 'zpow_n63' k7 'zpow_n64' k8 'zpow_n65'
 MTB > print k2 k6-k8
 
 signpow  0.307577
 zpow_n63 0.306073
 zpow_n64 0.308278
 zpow_n65 0.310472

 MTB > # Using n = 64 results in the power being closest to the power of the sign test.