Solution to HW 5 (power of correlation test)

> # Calculation for rho=0.30
>
> n<-seq(20,60)
> n
 [1] 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
[27] 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
> tstar<-sqrt(n-2)*.30/sqrt(1-.30^2)
> qt(.975,n-2)
 [1] 2.100922 2.093024 2.085963 2.079614 2.073873 2.068658 2.063899 2.059539
 [9] 2.055529 2.051831 2.048407 2.045230 2.042272 2.039513 2.036933 2.034515
[17] 2.032245 2.030108 2.028094 2.026192 2.024394 2.022691 2.021075 2.019541
[25] 2.018082 2.016692 2.015368 2.014103 2.012896 2.011741 2.010635 2.009575
[33] 2.008559 2.007584 2.006647 2.005746 2.004879 2.004045 2.003241 2.002465
[41] 2.001717
> cbind(n,tstar,qt(.975,n-2),tstar>qt(.975,n-2))
       n    tstar           
 [1,] 20 1.334249 2.100922 0
 [2,] 21 1.370810 2.093024 0
 [3,] 22 1.406422 2.085963 0
 [4,] 23 1.441153 2.079614 0
 [5,] 24 1.475068 2.073873 0
 [6,] 25 1.508219 2.068658 0
 [7,] 26 1.540658 2.063899 0
 [8,] 27 1.572427 2.059539 0
 [9,] 28 1.603567 2.055529 0
[10,] 29 1.634114 2.051831 0
[11,] 30 1.664101 2.048407 0
[12,] 31 1.693556 2.045230 0
[13,] 32 1.722508 2.042272 0
[14,] 33 1.750981 2.039513 0
[15,] 34 1.778998 2.036933 0
[16,] 35 1.806581 2.034515 0
[17,] 36 1.833750 2.032245 0
[18,] 37 1.860521 2.030108 0
[19,] 38 1.886913 2.028094 0
[20,] 39 1.912940 2.026192 0
[21,] 40 1.938619 2.024394 0
[22,] 41 1.963961 2.022691 0
[23,] 42 1.988981 2.021075 0
[24,] 43 2.013689 2.019541 0
[25,] 44 2.038099 2.018082 1
[26,] 45 2.062219 2.016692 1
[27,] 46 2.086060 2.015368 1
[28,] 47 2.109633 2.014103 1
[29,] 48 2.132944 2.012896 1
[30,] 49 2.156004 2.011741 1
[31,] 50 2.178819 2.010635 1
[32,] 51 2.201398 2.009575 1
[33,] 52 2.223748 2.008559 1
[34,] 53 2.245875 2.007584 1
[35,] 54 2.267787 2.006647 1
[36,] 55 2.289489 2.005746 1
[37,] 56 2.310987 2.004879 1
[38,] 57 2.332287 2.004045 1
[39,] 58 2.353394 2.003241 1
[40,] 59 2.374313 2.002465 1
[41,] 60 2.395050 2.001717 1
>
> # Calculation for rho=0.10
>
> n<-seq(350,400)
> tstar<-sqrt(n-2)*.10/sqrt(1-.10^2)
> cbind(n,tstar,qt(.975,n-2),tstar>qt(.975,n-2))
        n    tstar           
 [1,] 350 1.874874 1.966804 0
 [2,] 351 1.877566 1.966785 0
 [3,] 352 1.880254 1.966765 0
 [4,] 353 1.882938 1.966746 0
 [5,] 354 1.885618 1.966726 0
 [6,] 355 1.888295 1.966707 0
 [7,] 356 1.890967 1.966688 0
 [8,] 357 1.893636 1.966669 0
 [9,] 358 1.896302 1.966650 0
[10,] 359 1.898963 1.966631 0
[11,] 360 1.901621 1.966613 0
[12,] 361 1.904275 1.966594 0
[13,] 362 1.906925 1.966575 0
[14,] 363 1.909572 1.966557 0
[15,] 364 1.912215 1.966539 0
[16,] 365 1.914854 1.966521 0
[17,] 366 1.917490 1.966503 0
[18,] 367 1.920122 1.966485 0
[19,] 368 1.922751 1.966467 0
[20,] 369 1.925375 1.966449 0
[21,] 370 1.927997 1.966431 0
[22,] 371 1.930615 1.966414 0
[23,] 372 1.933229 1.966396 0
[24,] 373 1.935840 1.966379 0
[25,] 374 1.938447 1.966362 0
[26,] 375 1.941050 1.966344 0
[27,] 376 1.943651 1.966327 0
[28,] 377 1.946247 1.966310 0
[29,] 378 1.948841 1.966293 0
[30,] 379 1.951430 1.966276 0
[31,] 380 1.954017 1.966260 0
[32,] 381 1.956600 1.966243 0
[33,] 382 1.959179 1.966226 0
[34,] 383 1.961756 1.966210 0
[35,] 384 1.964328 1.966194 0
[36,] 385 1.966898 1.966177 1
[37,] 386 1.969464 1.966161 1
[38,] 387 1.972027 1.966145 1
[39,] 388 1.974586 1.966129 1
[40,] 389 1.977142 1.966113 1
[41,] 390 1.979695 1.966097 1
[42,] 391 1.982244 1.966081 1
[43,] 392 1.984791 1.966065 1
[44,] 393 1.987334 1.966050 1
[45,] 394 1.989873 1.966034 1
[46,] 395 1.992410 1.966019 1
[47,] 396 1.994943 1.966003 1
[48,] 397 1.997473 1.965988 1
[49,] 398 2.000000 1.965973 1
[50,] 399 2.002524 1.965957 1
[51,] 400 2.005044 1.965942 1
>
>  # If t_.975_n-2 is set to 2.0, then n is a simple function of rho
>
> r<-seq(.05,.95,.05)
> r
 [1] 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
0.75
[16] 0.80 0.85 0.90 0.95
> n<-4/r^2 - 2
> cbind(r,n)
         r           n
 [1,] 0.05 1598.000000
 [2,] 0.10  398.000000
 [3,] 0.15  175.777778
 [4,] 0.20   98.000000
 [5,] 0.25   62.000000
 [6,] 0.30   42.444444
 [7,] 0.35   30.653061
 [8,] 0.40   23.000000
 [9,] 0.45   17.753086
[10,] 0.50   14.000000
[11,] 0.55   11.223140
[12,] 0.60    9.111111
[13,] 0.65    7.467456
[14,] 0.70    6.163265
[15,] 0.75    5.111111
[16,] 0.80    4.250000
[17,] 0.85    3.536332
[18,] 0.90    2.938272
[19,] 0.95    2.432133