====== Wnioskowanie Statystyczne - Lista 8. ======
<code haskell>
leng = (\xs -> foldl (\x y -> x+1) 0 xs) 

ex xs = (sum xs) / (leng xs)

stddev xs = sqrt $ (/(leng xs)) $sum $ map (\x -> (x - (ex xs))^2) xs</code>


===== Zadanie 1. =====
===== Zadanie 2. =====
===== Zadanie 3. =====

  mx: 0.777000, my: 0.701000, sx: 1.462176, sy: 1.324404, cov: 0.383190, cor: 0.197876
  Prosta regresji: y = 0.561737x + 0.179232

<code python>
from itertools import imap, product
from math import sqrt
coords = [3, 3.5, 4, 4.5, 5]
fxy = {3:{}, 3.5:{}, 4:{}, 4.5:{}, 5:{}}

points = (
    (3, 3, 0.16),
    (3, 3.5, 0.12),
    (3, 4, 0.08),
    (3.5, 3.5, 0.04),
    (3.5, 4, 0.08),
    (3.5, 4.5, 0.12),
    (4, 4, 0.08),
    (4, 4.5, 0.06),
    (4, 5, 0.01),
    (4.5, 4.5, 0.05),
    (4.5, 5, 0.08),
    (5, 5, 0.08),
)

for x, y, v in points:
    fxy[x][y] = v

def f_xy((x, y)):
    return fxy[x].get(y, 0)

def fx(y):
    fx = 0
    for x in coords:
        fx += fxy[x].get(y, 0)
    return fx

def fy(x):
    fy = 0
    for y in coords:
        fy += fxy[x].get(y, 0)
    return fy

def idty(x):
    return x

def E(f, g, cs):
    ls = [g(c)*f(c) for c in cs]
    return sum(ls)/len(ls) 

mx = E(fx, idty, coords)
my = E(fy, idty, coords)

sx = sqrt(E(fx, lambda x: (x-mx)*(x-mx), coords))
sy = sqrt(E(fy, lambda y: (y-my)*(y-my), coords))

cov = E(f_xy, lambda (x, y): (x-mx)*(y-my), product(coords, repeat=2))
cor = cov/(sx*sy)

print "mx: %f, my: %f, sx: %f, sy: %f, cov: %f, cor: %f" % (mx, my, sx, sy, cov, cor)

# E([Y-(a+bX)]^2) - min

# b = CXY / D^2X, a = EY - bEX

b = cov / (sx*sx)
a = my - b*mx

print "Prosta regresji: y = %fx + %f" % (a, b)

</code>

===== Zadanie 4. =====
===== Zadanie 5. =====
Mała próbka.\\
$\overline X = 333$\\
$S = 33.16323265304515$\\
$\displaystyle P \left( \overline{X} - t_{1 - \frac{\alpha}{2}} \frac{S}{\sqrt{n-1}} < m < \overline{X} + t_{1 - \frac{\alpha}{2}} \frac{S}{\sqrt{n-1}} \right) = 1 - \alpha$
===== Zadanie 6. =====
Mała próbka.\\
$\overline X = 1.574$\\
$S = 0.043$\\
$\displaystyle P \left( \overline{X} - t_{1 - \frac{\alpha}{2}} \frac{S}{\sqrt{n-1}} < m < \overline{X} + t_{1 - \frac{\alpha}{2}} \frac{S}{\sqrt{n-1}} \right) = 1 - \alpha$
===== Zadanie 7. =====
Duża próbka.\\
$\overline X = 37.5$\\
$S = 39.566399886772615$\\
$\displaystyle P \left( \overline{X} - u_{1-\frac{\alpha}{2}} \frac{S}{\sqrt{n}} < m < \overline{X} + u_{1-\frac{\alpha}{2}} 
 \frac{S}{\sqrt{n}} \right) = 1 - \alpha$
===== Zadanie 8. =====
Mała próbka.\\
$\overline X = 100.0$\\
$S = 11.586630226256467$\\
To jest dla wariancji, czyli pierwiastek, z obu stron:\\
$\displaystyle P \left( \frac{nS^2}{\chi^{2}_{1 - \frac{\alpha}{2}, n - 1}} < \sigma^2 < \frac{nS^2}{\chi^{2}_{\frac{\alpha}{2}, n - 1}} \right)= 1 - \alpha$
===== Zadanie 9. =====
Duża próbka.\\
$\displaystyle P \left( \frac{S}{1+ \frac{u_{\alpha}}{\sqrt{2n}}} < \sigma < \frac{S}{1 - \frac{u_{\alpha}}{\sqrt{2n}}} \right)= 1 - \alpha$
===== Zadanie 10. =====
Duża próbka.

Przedział:

$p = \frac k n$

$\displaystyle \left(p - u_{1-\alpha/2} \sqrt{\frac {p(1-p)}n}, p + u_{1-\alpha/2} \sqrt{\frac {p(1-p)}n}\right)$


===== Zadania 5-10 - wyliczenie =====
<code>
5.
[(312.96431003654271, 353.03568996345729), (308.36923762299926, 357.63076237700074), (297.96555773250361, 368.03444226749639)]

6.
[(-9.7387607780359318, 12.886760778035931), (-12.059600199779362, 15.207600199779362), (-16.856231309036904, 20.004231309036907)]

7.
[(33.742550387392015, 41.257449612607985), (33.022722633937505, 41.977277366062495), (31.615860122640775, 43.384139877359225)]

8.
[(9.2700847290948065, 17.199689112976362), (8.8398005031199745, 18.520633113068985), (8.0922906651952005, 21.607026599096702)]

9.
[(0.022687893099167609, 0.03178546605420815), (0.022082482141430701, 0.033055090136681231), (0.020987900459555682, 0.035854128204083208)]

10.
[(0.089306808583436648, 0.19402652474989668), (0.079276047588324225, 0.20405728574500909), (0.059671494226666497, 0.22366183910666682)]
</code>
<code python>
from scipy.stats import t, norm, chi2
from math import sqrt

n = 10
m = 333
S = 33.16323265304515

conf_t = map(lambda x: t.ppf(1-x/2, n), [0.1, 0.05, 0.01])

print "5."
print map(lambda conf: (m-conf*S/sqrt(n-1), m+conf*S/sqrt(n-1)), conf_t)
print

n = 26
m = 1.574
s = 0.043

conf_t = map(lambda x: t.ppf(1-x/2, n), [0.1, 0.05, 0.01])

print "6."
print map(lambda conf: (m-conf*S/sqrt(n-1), m+conf*S/sqrt(n-1)), conf_t)
print

n = 300
m = 37.5
S = 39.566399886772615

conf_u = map(lambda x: norm.ppf(1-x/2), [0.1, 0.05, 0.01])

print "7."
print map(lambda conf: (m-conf*S/sqrt(n), m+conf*S/sqrt(n)), conf_u)
print

n = 16
m = 100
S = 11.586630226256467

conf_chi = map(lambda a: (chi2.ppf(1-a/2, n-1), chi2.ppf(a/2,n-1)), [0.1, 0.05, 0.01])

print "8."
print map(lambda (conf1, conf2): (sqrt(n*S*S/conf1), sqrt(n*S*S/conf2)), conf_chi)
print

n = 50
S = sqrt(0.00068)

conf_u = map(lambda x: norm.ppf(1-x/2), [0.1, 0.05, 0.01])

print "9."
print map(lambda conf: (S*sqrt(2*n)/(sqrt(2*n-3)+conf),S*sqrt(2*n)/(sqrt(2*n-3)-conf)), conf_u)
print 


n = 120.0
k = 17.0
p = k/n

print "10."
print map(lambda conf: (p - conf*sqrt(p*(1-p)/n), p + conf*sqrt(p*(1-p)/n)), conf_u)
print 
</code>