Introduction

In the world of statistical analysis, both hypothesis tests and confidence intervals play crucial roles in drawing conclusions about populations based on sample data. Understanding the relationship between these two concepts is essential for making informed decisions and interpreting results accurately. In this chapter, we will delve into how hypothesis tests and confidence intervals are interconnected, and how they provide complementary insights into the parameters of interest.

Interplay Between Hypothesis Tests and Confidence Intervals:

Hypothesis testing and confidence intervals are two distinct approaches to statistical inference, but they share a close relationship. While hypothesis testing focuses on assessing whether a particular claim about a population parameter is supported by the data, confidence intervals provide a range of plausible values for the parameter.

In essence, a confidence interval is a range of values within which we have a certain level of confidence that the true population parameter lies. This level of confidence is often denoted as 1-$\alpha$, where $\alpha$ is the chosen level of significance. For example, a 95% confidence interval implies that we are 95% confident that the true parameter falls within the interval.

The formula for a confidence interval for a population mean $(\mu)$ when the population standard deviation $(\sigma)$ is known is:

$$\text{Confidence Interval}=\bar{x} \pm z\left(\frac{\sigma}{\sqrt{n}}\right)$$ where

• $\bar{x}$ is the sample mean
• $z$ is the critical value from the standard normal distribution
• $\sigma$ is the population standard deviation
• $n$ is the sample size

Now, how does this relate to hypothesis testing? Consider a scenario where we are testing the null hypothesis that the population mean $(\mu)$ equals a specific value $\left(\mu_0\right)$. If we construct a $95 \%$ confidence interval for $\mu$ and find that the null hypothesis value $\left(\mu_0\right)$ is within this interval, we fail to reject the null hypothesis. On the other hand, if the null hypothesis value falls outside the confidence interval, we might have evidence to reject the null hypothesis in favor of an alternative claim.