A t-test is any statistical hypothesis test in which the test statistic ...
Student's t-test
A t-test is any statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is true.
Student's t-distribution
Suppose X1, X2, ..., Xn are independent random variables that are normally distributed with mean ? and variance 2.
sample mean: X n= ( X1 + ... + X n ) / n
sample va= riance: Sn2
1 n -1
n i =1
(Xi
-
Xn )2
T X= n - ? Z X n - ?
Sn n
n
Z is normally distributed with mean 0 and variance 1 T has a Student's t-distribution with n-1 degrees of freedom.
The t-distribution looks like the standard normal distribution (exact with n+) with fatter tails.
Independent one-sample t-test
Independent one-sample t-test:
Null hypothesis: the population mean is equal to a specified value ?0.
Test static: t = x - ?0
sn
s: the sample standard deviation n: sample size Degree of freedom (d.o.f) = n-1
Independent two-sample t-test
1): Equal sample size, equal variance
Null hypothesis: the population mean is equal.
Test static:
t = X1 - X2
S X1X 2
2 n
where SX1X2 =
S2 X1
+
SX2
2
2
? SX1,SX2: the sample standard deviation from each group.
? n: participants of each group ? Degree of freedom (d.o.f) = 2n-2
Independent two-sample t-test
2): Unequal sample size, equal variance
Null hypothesis: the population mean is equal.
Test static:
t = X1 - X2
S X1X 2
1+1 n1 n2
where SX1X2 =
(n1 -1)SX12 + (n2 - 2)SX2 2 n1 + n2 - 2
? SX1,SX2: the sample standard deviation from each group.
? n: participants of each group ? Degree of freedom (d.o.f) = 2n-2
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