The document discusses Student's t-test and its applications. It begins by defining a Student's t-test as a statistical hypothesis test where the test statistic follows a Student's t-distribution if the null hypothesis is true. It then describes how the Student's t-distribution arises from independent, normally distributed random variables. Finally, it outlines three common types of t-tests: the independent one-sample t-test to compare a sample mean to a hypothesized population mean; the independent two-sample t-test with equal sample sizes and variances; and the independent two-sample t-test with unequal sample sizes but equal variances.

1. 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.

2. Student’s t-distribution
Suppose X1, X2, …, Xn are independent random variables that are normally
distributed with mean µ and variance σ2.
1
2 2
1
sample mean: ( ... ) /
1
sample variance: ( )
1
Z
n n
n
n i n
i
n n
n
X X X n
S X X
n
X X
T
S n n
µ µ
σ
=
= + +
= −
−
− −
=
∑
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.

3. Independent one-sample t-test
Independent one-sample t-test:
Null hypothesis: the population mean is equal to a specified value µ0.
Test static:
0
x
t
s n
µ
−
=
s: the sample standard deviation
n: sample size
Degree of freedom (d.o.f) = n-1

4. Independent two-sample t-test
1): Equal sample size, equal variance
Null hypothesis: the population mean is equal.
Test static:
1 2
1 2
1 2
1 2
2 2
2
where
2
X X
X X
X X
X X
t
S
n
S S
S
−
=
+
=
• SX1,SX2: the sample standard deviation from
each group.
• n: participants of each group
• Degree of freedom (d.o.f) = 2n-2

5. Independent two-sample t-test
2): Unequal sample size, equal variance
Null hypothesis: the population mean is equal.
Test static:
1 2
1 2
1 2
1 2
1 2
2 2
1 2
1 2
1 1
( 1) ( 2)
where
2
X X
X X
X X
X X
t
S
n n
n S n S
S
n n
−
=
+
− + −
=
+ −
• SX1,SX2: the sample standard deviation
from each group.
• n: participants of each group
• Degree of freedom (d.o.f) = 2n-2