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

$$\bigg(\overline{X} -t_{(\alpha/2,n-1)} \dfrac{s}{\sqrt{n}}, \overline{X} +t_{(\alpha/2,n-1)} \dfrac{s}{\sqrt{n}}\bigg)$$

References & Further Reading

  • Anderson, D. R., Sweeney, D. J., & Williams, T. A. (2018). Statistics for Business and Economics (13th ed.). Cengage Learning.
  • Kline, R. B. (2013). Becoming a Behavioral Science Researcher: A Guide to Producing Research that Matters. Guilford Press.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.
  • National Institute of Standards and Technology (NIST). Engineering Statistics Handbook - Confidence Intervals section
  • R Documentation: t.test() function for computing t-based confidence intervals

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