In this tutorial we will discuss paired $t$test, assumptions for paired $t$test and step by step procedure for paired $t$test. Paired $t$ test is also known as dependent sample ttest.
Paired ttest dependent sample
Let $X_1, X_2, \cdots, X_n$ and $Y_1, Y_2, \cdots, Y_n$ be two dependent samples each of size $n$ with respective means $\mu_1$ and $\mu_2$.
Define $d_i = X_i – Y_i$, $i=1,2,\cdots, n$. Then $\mu_d= \mu_1 – \mu_2$.
Let $\overline{d}=\frac{1}{n} \sum d_i$ be the mean of the difference and $s_d=\sqrt{\frac{1}{n1}\sum (d_i – \overline{d})^2}$ be the sample standard deviation of the difference.
Assumptions
a. The samples are dependent (matched pairs).
b. Both the samples are simple random sample.
c. The two samples are both large ($n_1 > 30$ and $n_2 >30$) or both the samples comes from population having normal distribution.
Step by Step Procedure
We wish to test the hypothesis $H_0: \mu_1 = \mu_2$, i.e., $H_0:\mu_d=0$.
The standard error of $\overline{d}$ is
$$ \begin{aligned} SE(\overline{d}) = \frac{s_d}{\sqrt{n}} \end{aligned} $$
where $s_d$ is the sample standard deviation of the difference $d_i$.
Step 1 State the hypothesis testing problem
The hypothesis testing problem can be structured in any one of the three situations as follows:
Situation  Hypothesis Testing Problem 

Situation A  $H_0: \mu_d=0$ against $H_a : \mu_d < 0$ (Lefttailed) 
Situation B  $H_0: \mu_d=0$ against $H_a : \mu_d > 0$ (Righttailed) 
Situation C  $H_0: \mu_d=0$ against $H_a : \mu_d \neq 0$ (Twotailed) 
Step 2 Define the test statistic
The test statistic for testing above hypothesis is
$$ \begin{eqnarray*} t & =& \frac{\overline{d}\mu_d}{SE(\overline{d})}\\ & =& \frac{\overline{d}\mu_d}{s_d/\sqrt{n}} \end{eqnarray*} $$
where $\overline{d} =\frac{1}{n}\sum d_i$ and $s_d=\sqrt{\frac{\sum (d_i \overline{d})^2}{n1}}$.
The test statistic defined above follows Students $t$ distribution with $n1$ degrees of freedom.
Step 3 Specify the level of significance $\alpha$.
Step 4 Determine the critical values
For the specified value of $\alpha$ determine the critical region depending upon the alternative hypothesis.
 For lefttailed alternative hypothesis: Find the $t$critical value using
$$ \begin{aligned} P(t<t_\alpha) = \alpha. \end{aligned} $$
 For righttailed alternative hypothesis: $t_\alpha$.
$$ \begin{aligned} P(t>t_\alpha) = \alpha. \end{aligned} $$
 For twotailed alternative hypothesis: $t_{\alpha/2}$.
$$ P(t<t_{\alpha/2} \text{ or } t> t_{\alpha/2}) = \alpha. $$
Step 5 Computation
Compute the test statistic under the null hypothesis $H_0$ using equation
$$ \begin{aligned} t_{obs} & = \frac{\overline{d}0}{s_d/\sqrt{n}} \end{aligned} $$
Step 6 Decision (Traditional Approach)
It is based on the critical values.

For lefttailed alternative hypothesis: Reject $H_0$ if
$t_{obs}\leq t_\alpha$
. 
For righttailed alternative hypothesis: Reject $H_0$ if
$t_{obs}\geq t_\alpha$
. 
For twotailed alternative hypothesis: Reject $H_0$ if
$t_{obs}\geq t_{\alpha/2}$
.
OR
Step 6 Decision ($p$value Approach)
It is based on the $p$value.
Alternative Hypothesis  Type of Hypothesis  $p$value 

$H_a: \mu_1<\mu_2$  Lefttailed  $p$value $= P(t\leq t_{obs})$ 
$H_a: \mu_1>\mu_2$  Righttailed  $p$value $= P(t\geq t_{obs})$ 
$H_a: \mu_1\neq \mu_2$  Twotailed  $p$value $= 2P(t\geq abs(t_{obs}))$ 
If $p$value is less than $\alpha$, then reject the null hypothesis $H_0$ at $\alpha$ level of significance, otherwise fail to reject $H_0$ at $\alpha$ level of significance.
Endnote
In this tutorial, you learned the paired $t$test and the assumptions for paired $t$test. You also learned about the step by step procedure to apply paired $t$test.
To learn more about other hypothesis testing problems, hypothesis testing calculators and step by step procedure, please refer to the following tutorials:
Let me know in the comments if you have any questions on paired $t$test and your thought on this article.