Introduction
Confidence interval can be used to estimate the population parameter with the help of an interval with some degree of confidence. One such a parameter that can be estimated is a population mean.
In this article we will discuss step by step procedure to construct a confidence interval for population mean when the population standard deviation is unknown.
Confidence Interval for the mean ($\sigma$ unknown)
Let $X_1, X_2, \cdots , X_{n}$ be a random sample of size $n$ from $N(\mu, \sigma^2)$ with unknown variance $\sigma^2$.
Let $\overline{X} = \frac{1}{n} \sum X_i$ be the sample mean. Let $s =\sqrt{\frac{1}{n-1}\sum (X_i - \overline{X})^2}$ be the sample standard deviation.
Let $\alpha$ be the level of significance. Then $(1-\alpha)$ is called the confidence coefficient. We wish to construct a $100(1-\alpha)$% confidence interval of a population mean $\mu$ when $\sigma$ is unknown.
The margin of error for mean is
$$ \begin{aligned} E = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}. \end{aligned} $$
Then, $100(1-\alpha)$% confidence interval for population mean (when $\sigma$ unknown) is
$$ \begin{aligned} \overline{X} - E \leq \mu \leq \overline{X} + E. \end{aligned} $$
Assumptions
a. The sample is a simple random sample.
b. The population standard deviation $\sigma$ is unknown.
c. The population is normally distributed or $n<30$.
Step by Step Procedure
Step by step procedure to estimate the confidence interval for mean is as follows:
Step 1 Specify the confidence level $(1-\alpha)$
Step 2 Given information
Specify the given information, sample size $n$, sample mean $\overline{X}$ and sample standard deviation $s$.
Step 3 Specify the formula
$100(1-\alpha)$% confidence interval for the population mean $\mu$ is
$$ \begin{aligned} \overline{X} - E \leq \mu \leq \overline{X} + E \end{aligned} $$
where $E = Z_{\alpha/2} \frac{s}{\sqrt{n}}$.
Step 4 Determine the critical value
Determine the critical value $t_{(\alpha/2,n-1)}$ from $t$ statistical table that corresponds to the desired confidence level and the degrees of freedom.
Step 5 Compute the margin of error
The margin of error for mean is
$$ \begin{aligned} E = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}. \end{aligned} $$
Step 6 Determine the confidence interval
$100(1-\alpha)$% confidence interval estimate for population mean is
$$ \begin{aligned} \overline{X} - E \leq \mu\leq \overline{X} + E \end{aligned} $$
Equivalently, $100(1-\alpha)$% confidence interval estimate of population mean is $\overline{X} \pm E$ or $(\overline{X} -E, \overline{X} +E)$.
That is $100(1-\alpha)$% confidence interval estimate of population mean (when $\sigma$ unknown) is
$$\bigg(\overline{X} -t_{(\alpha/2,n-1)} \dfrac{s}{\sqrt{n}}, \overline{X} +t_{(\alpha/2,n-1)} \dfrac{s}{\sqrt{n}}\bigg)$$
