# Sample means in the Planet X

Discussion in 'Scientific Statistics Math' started by Luis A. Afonso, Dec 3, 2010.

1. ### Luis A. AfonsoGuest

Sample means in the Planet X

Since one is able to simulate the Universe itâ€™s a straightforward duty to obtain a Critical Values of a Test Statistics of samples draw from it. This fact, at least 50 years old (see Lilliefors Test, 1968) is not known by Jack Tomsky yet. I several times, since 2007, ask him to reformulate his ideas about the Monte Carlo Method but had not got any feedback.
A simple exercise concerning X~N(0,1): 10 samples and the statistics t = sqrt(n) * (xhat - 0)/ s involving the sample mean and its standard deviation is quite explicit.

The quantiles of t were (1 million samples):
_____0.60___0.260____Tables____0.261
_____0.75___0.703_____________ 0.703
_____0.90___1.383_____________ 1.383
_____0.95___1.832_____________ 1.833
_____0.975__2.258_____________ 2.262
_____0.99___2.827_____________ 2.821

Itâ€™s conceivable that in a planet where Gasset/Student had never landed, their inhabitants, though sufficiently experts in Microelectronics, could fully be able to find out confident intervals for normal sample means.

Luis

REM "MATE"
CLS
PRINT
PRINT " T=sqr(n)*(Xhat-0)/s fractiles samples N(0,1) "
pi = 4 * ATN(1)
INPUT " all = "; all
DIM w(8001)
LOCATE 8, 1
PRINT " 0.600 0.750 0.900 0.950 ";
PRINT " 0.975 0.990 "
FOR n = 5 TO 20
PRINT : COLOR 14
FOR j = 0 TO 8001: w(j) = 0: NEXT j
FOR j = 1 TO all
LOCATE 5, 50
PRINT USING "##########"; all - j
RANDOMIZE TIMER
m = 0: xx = 0
FOR i = 1 TO n
aa = SQR(-2 * LOG(RND))
x = 1 * aa * COS(2 * pi * RND)
m = m + x / n: xx = xx + x * x
NEXT i
v = xx - n * m * m: v = v / (n - 1)
t = m / (SQR(v) / SQR(n))
IF t < -4 THEN t = -4
IF t > 4 THEN t = 4
t = t + 4
t = INT(1000 * t + .5)
w(t) = w(t) + 1 / all
NEXT j: PRINT : PRINT : PRINT
LOCATE 5 + n, 1
PRINT USING "###"; n;
v(1) = .6: v(2) = .75: v(3) = .9: v(4) = .95
v(5) = .975: v(6) = .99
FOR vv = 1 TO 6: s = 0
FOR tt = 0 TO 8000
s = s + w(tt)
IF s > v(vv) THEN GOTO 10
NEXT tt
10 PRINT USING " ##.### "; tt / 1000 - 4;
NEXT vv
NEXT n
END

Luis A. Afonso, Dec 3, 2010