Dear Sir
I will try to explain your other questions.
Helas, the tables of your example are unreadable to me.
I am not a specialist in statistics, but I made a research in my books
that I used during my studies at the university of Gent(Belgium).
The distribution function that you need to use is dependent of the
distribution of the results that you use. Therefore you have to study
the graph and compare it with the typical graphs of the distribution
functions. Also its sometimes possible to do it with software.
If you do a hypothesis, we can distinguish the following situations:
1)Test: ONE population, sigma^2 known, normal average
a)H0(= Hypothesis 0): µ = µ0
b)Choose an alpha (f.e. 0.5)
c)Calculate Xn en Z = (Xn - µ0)/(sigma/root(n))
d)choose an alternative hypothesis
d.1)H1:µ < µ0
reject H0 if Z < Phi^(-1)(alpha)
d.2)H1:µ > µ0
reject H0 if Z > Phi^(-1)(1-alpha)
d.3)H1:µ <> µ0
reject H0 if |Z| > Phi^(-1)(1-alpha/2)
2)T-test: ONE population, sigma^2 unknown, normal average
a)H0(= Hypothesis 0): µ = µ0
b)Choose an alpha (f.e. 0.5)
c)Calculate Xn , Sn^2 en T = (Xn - µ0)/root(Sn^2/n)
d)choose an alternative hypothesis
d.1)H1:µ < µ0
reject H0 if T < t(n-1)^(-1)(alpha)
d.2)H1:µ > µ0
reject H0 if T > t(n-1)^(-1)(1-alpha)
d.3)H1:µ <> µ0
reject H0 if |T| > t(n-1)^(-1)(1-alpha/2)
3)Xhi^2 test: ONE population, µ known, normal average
a)H0(= Hypothesis 0): sigma^2 = sigma0^2
b)Choose an alpha (f.e. 0.5)
c)Calculate sigma^2 = SUM(i=1,n)(Xi - µ)^2)/n and X^2 = n * sigma^2/sigma0^2
d)choose an alternative hypothesis
d.1)H1: sigma^2 < sigma0^2
reject H0 if Xhi^2< (Xhi_n^2)^(-1)(alpha)
d.2)H1: sigma^2 < sigma0^2
reject H0 if Xhi^2 > (Xhi_n^2)^(-1)(1-alpha)
d.3)H1: sigma^2 <> sigma0^2
reject H0 if Xhi^2 > (Xhi_n^2)^(-1)(1-alpha/2)
OR Xhi^2 < (Xhi_n^2)^(-1)(alpha/2)
4)Xhi^2 test: ONE population, µ unknown, normal average
a)H0(= Hypothesis 0): sigma^2 = sigma0^2
b)Choose an alpha (f.e. 0.5)
c)Calculate Xn,Sn^2 and X^2 = (n-1) * Sn^2/sigma0^2
d)choose an alternative hypothesis
d.1)H1: sigma^2 < sigma0^2
reject H0 if Xhi^2< (Xhi_(n-1)^2)^(-1)(alpha)
d.2)H1: sigma^2 < sigma0^2
reject H0 if Xhi^2 > (Xhi_(n-1)^2)^(-1)(1-alpha)
d.3)H1: sigma^2 <> sigma0^2
reject H0 if Xhi^2 > (Xhi_(n-1)^2)^(-1)(1-alpha/2)
OR Xhi^2 < (Xhi_(n-1)^2)^(-1)(alpha/2)
5)F(isher) test: 2 populations, µ1 and µ2 known, average variances
a)H0(= Hypothesis 0): sigma1^2 = sigma2^2
b)Choose an alpha (f.e. 0.5)
c)Calculate sigma1^2=(SUM(j=1,n)(Xj-µ1)^2)/n
, sigma2^2=(SUM(j=1,m)(Yj-µ2)^2)/m AND F = sigma1^2/sigma2^2
d)choose an alternative hypothesis
d.1)H1: sigma1^2 < sigma2^2
reject H0 if F < F(n,m)^(-1)(alpha)
d.2)H1: sigma1^2 < sigma2^2
reject H0 if F > F(n,m)^(-1)(1-alpha)
d.3)H1: sigma1^2 <> sigma2^2
reject H0 if F > F(n,m)^(-1)(1-alpha/2)
OR F < F(n,m)^(-1)(alpha/2)
6)F(isher) test: 2 populations, µ1 and µ2 unknown, average variances
a)H0(= Hypothesis 0): sigma1^2 = sigma2^2
b)Choose an alpha (f.e. 0.5)
c)Calculate S1^2,S2^2,AND F = S1^2/S2^2
d)choose an alternative hypothesis
d.1)H1: sigma1^2 < sigma2^2
reject H0 if F < F(n-1,m-1)^(-1)(alpha)
d.2)H1: sigma1^2 < sigma2^2
reject H0 if F > F(n-1,m-1)^(-1)(1-alpha)
d.3)H1: sigma1^2 <> sigma2^2
reject H0 if F > F(n-1,m-1)^(-1)(1-alpha/2)
OR F < F(n-1,m-1)^(-1)(alpha/2)
7)z-test: 2 populations, sigma1^2 and sigma2^2 known, normal averages
a)H0(= Hypothesis 0): µ1 - µ2 = delta
b)Choose an alpha (f.e. 0.5)
c)Calculate Xn, Ym and Z = (Xn - Ym - delta) / Root(sigma1^2 /n +sigma2^2 /m)
d)choose an alternative hypothesis
d.1)H1: µ1 - µ2 < delta
reject H0 if Z < Phi^-1(alpha)
d.2)H1: µ1 - µ2 > delta
reject H0 if Z > Phi^-1(1-alpha)
d.3)H1: µ1 - µ2 <> delta
reject H0 if |Z| > Phi^-1(1-alpha/2)
8)t-test: 2 populations, sigma1^2 and sigma2^2 unknown, normal averages
a)H0(= Hypothesis 0): µ1 - µ2 = delta
b)Choose an alpha (f.e. 0.5)
c)Calculate Xn, Ym, S1^2, S2^2, Sp^2 = [(n-1)S1^2+(m-1S2^2]/(n+m-2)
and T = (Xn - Ym -delta)/root(Sp^2(n+m)/(nm))
d)choose an alternative hypothesis
d.1)H1: µ1 - µ2 < delta
reject H0 if T < T(n+m-2)^-1(alpha)
d.2)H1: µ1 - µ2 > delta
reject H0 if T > T(n+m-2)^-1(1-alpha)
d.3)H1: µ1 - µ2 <> delta
reject H0 if |T| > T(n+m-2)^-1(1-alpha/2)
I hope this can satisfy your question and you will give me the
permission to post the answers as an official one.
If not, i am afraid i cant help you anymore.
Kind Regards
Pythagoras |
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