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 known.

Confidence Interval for Mean

Let $X_1, X_2, \cdots , X_{n}$ be a random sample of size $n$ from $N(\mu, \sigma^2)$ with known variance $\sigma^2$. Let $\overline{X} = \frac{1}{n} \sum X_i$ be the sample mean.

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 known.

The margin of error for mean is $$ \begin{aligned} E = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}. \end{aligned} $$ Then, $100(1-\alpha)$% confidence interval for the population mean $\mu$ 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 known.

c. The population has a normal distribution.

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 population standard deviation.

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 $E = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$.

Step 4 Determine the critical value

Determine the critical value $Z_{\alpha/2}$ from normal statistical table that corresponds to the desired confidence level.

Step 5 Compute the margin of error

The margin of error for mean is $$ \begin{aligned} E = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}. \end{aligned} $$

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 the $100(1-\alpha)$% confidence interval estimate of population mean (when $\sigma$ is unknown) is

$$\bigg(\overline{X} -Z_{\alpha/2} \dfrac{\sigma}{\sqrt{n}}, \overline{X} +Z_{\alpha/2} \dfrac{\sigma}{\sqrt{n}}\bigg)$$

References

  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineers & Scientists (9th ed.). Pearson.
  • Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (6th ed.). Wiley.
  • Kish, L. (1965). Survey Sampling. Wiley.