## Confidence Interval for paired t-test

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 $d_i = X_i - Y_i$, $i=1,2,\cdots, n$. Let $\overline{d}=\frac{1}{n} \sum d_i$ be the mean of the difference and $s_d=\sqrt{\frac{1}{n-1}\sum (d_i - \overline{d})^2}$ be the sample standard deviation of the difference.

Let $C=1-\alpha$ be the confidence coefficient. Our aim is to construct $100(1-\alpha)$% confidence interval estimate of the mean of the differences from dependent samples.

The margin of error is
```
$$
\begin{aligned}
E &= t_{(\alpha/2,n-1)} \frac{s_d}{\sqrt{n}}.
\end{aligned}
$$
```

where $t_{(\alpha/2,n-1)}$ is the value from $t$ statistical table for desired confidence level and degrees of freedom.

Then $100(1-\alpha)$% confidence interval for the mean of the difference is
```
$$
\begin{aligned}
& \overline{d} - E \leq \mu \leq \overline{d} + E.
\end{aligned}
$$
```

## Assumptions

a. The two 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

Step by step procedure to estimate the confidence interval for difference between two population means for dependent samples is as follows:

### Step 1 Specify the confidence level $(1-\alpha)$

### Step 2 Given information

Specify the given information, sample size $n$, sample mean value of the difference $\overline{d}$ and the sample standard deviation $s_d$.

### Step 3 Specify the formula

$100(1-\alpha)$% confidence interval for the mean of the difference is

```
$$
\begin{aligned}
\overline{d} - E \leq \mu \leq \overline{d} + E.
\end{aligned}
$$
```

where $E = t_{(\alpha/2,n-1)} \frac{s_d}{\sqrt{n}}$.

### Step 4 Determine the critical value

Determine the critical value $t_{(\alpha/2,n-1)}$ that corresponds to the desired confidence level.

### Step 5 Compute the margin of error

The margin of error is

```
$$
\begin{aligned}
E = t_{(\alpha/2,n-1)} \frac{s_d}{\sqrt{n}}.
\end{aligned}
$$
```

### Step 6 Determine the confidence interval

$100(1-\alpha)$% confidence interval estimate for the mean of the difference is

```
$$
\begin{aligned}
\overline{d} - E \leq \mu\leq \overline{d} + E
\end{aligned}
$$
```

Equivalently, $100(1-\alpha)$% confidence interval estimate of the mean of the difference is $\overline{d} \pm E$ or $(\overline{d} -E, \overline{d} +E)$.