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 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 $C=1-\alpha$ be 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 standrad 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
$(1-\alpha)*100\%$ 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
$$\bigg(\overline{X} -t_{(\alpha/2,n-1)} \dfrac{s}{\sqrt{n}}, \overline{X} +t_{(\alpha/2,n-1)} \dfrac{s}{\sqrt{n}}\bigg)$$.
Reference
You can read step by step tutorial on Confidence interval for mean when signma is unknown examples, tutorial will help you to understand how to construct confidence interval for population mean when the population standard deviation is unknown with example.
You can also use calculator to compute the confidence interval for population mean when the population standard deviation is unknown.
Confidence interval for mean when signma unknown examples
Confidence Interval for Mean (t) Calculator
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