Introduction

In the realm of hypothesis testing, constructing and applying confidence intervals is a fundamental skill that aids in making informed decisions about population parameters. Confidence intervals provide a range of values within which the true parameter value is likely to fall. In this chapter, we will explore the process of constructing confidence intervals for both one-sided and two-sided hypothesis tests, and learn how to interpret the results with a specific confidence level.

Confidence intervals serve as a powerful tool in hypothesis testing by offering a range of plausible values for a population parameter. Whether it’s a mean, proportion, or another parameter, confidence intervals provide us with a level of uncertainty surrounding the estimated value. This chapter will delve into the nuances of constructing and applying confidence intervals for various scenarios and hypothesis tests.

Confidence Intervals for One-Sided Tests

For one-sided hypothesis tests, where the researcher is interested in testing if a parameter is either greater or lesser than a certain value, the confidence interval can be tailored to reflect this focus. To construct a confidence interval for the lower or upper limit, we use the following formulas:

• For a lower one-sided interval:

$$\text{Lower Limit = Sample Mean} -( \text{Critical Value} \times \text{Standard Error} )$$

• For an upper one-sided interval: $$\text{Upper Limit = Sample Mean} + ( \text{Critical Value} \times \text{Standard Error})$$

Confidence Intervals for Two-Sided Tests

In the case of two-sided hypothesis tests, where we want to determine if a parameter is different from a specific value, the confidence interval involves capturing the potential range around the sample mean. The formula for a two-sided confidence interval is:

$$\text{Confidence Interval= Sample Mean}\pm ($ \text{Critical Value} \times \text{Standard Error})$$

Interpreting Results with a Specific Confidence Level

Interpreting the results of confidence intervals involves considering the chosen confidence level. For example, a 95% confidence interval implies that in repeated sampling, the true population parameter would fall within the interval 95% of the time. If the calculated confidence interval contains the hypothesized value, it suggests that the null hypothesis is plausible within the chosen confidence level.

Example: A pharmaceutical company claims that the average weight loss achieved by their new drug is at least 5 pounds. A random sample of 50 individuals who took the drug showed an average weight loss of 6 pounds with a standard deviation of 1.5 pounds. Using a 95\% confidence level, we can construct a one-sided confidence interval for the lower limit:

$$\text { Lower Limit }=6-\left(1.645 \times \frac{1.5}{\sqrt{50}}\right)$$

After calculations, we find the lower limit to be approximately 5.72 pounds. Since this interval does not contain the value 5 , it suggests that the claim of the company might be true within a $95 \%$ confidence level.