What is a Confidence Interval?

What is a Confidence Interval?

A confidence interval is a range of parameters that complements the rejection region. For example, a 95% this contains this set of parameter values where the null hypothesis cannot be rejected when using a 5% test. More generally, a 1 – α confidence interval contains the values that cannot be rejected when using a test size of .

The bounds of a confidence interval depend on the type of alternative used in the test.

Example of Confidence Interval:

When Ho:μ is tested against the two-sided alternative (e.g. H₁:μ is not equal to μ_o), then the 1 – α, the confidence interval is:

$ [ \mu \, -\, C_{\alpha }\, \times \frac{\sigma }{\sqrt{n},\mu }\, +\, C_{\alpha }\, \times \frac{\sigma }{\sqrt{n}} ] $

Where C_α is a critical value for a two-sided test (e.g. if α = 5%, then the C_α = 1.96)

Why This is important?

Confidence intervals are critical when generalizing data since they show the range of scores that are likely if the survey is repeated. It assists us in determining the level we are confident about the results.

It is also essential in hypothesis testing. They allow us to determine whether the null hypothesis should be accepted or rejected. By calculating a confidence interval, we can determine whether the hypothesized value falls within the range of values that are likely based on the sample data. If the hypothesized value falls within the confidence interval, we fail to reject the null hypothesis. If the hypothesized value falls outside the confidence interval, we reject the null hypothesis. It is also useful in communicating the uncertainty in data to non-experts, allowing them to better understand the results of a study or experiment.

This also plays an important role in hypothesis testing. In statistical hypothesis testing, a confidence interval is used to estimate the population parameter and determine if it falls within a certain range of values. If the calculated confidence interval includes the hypothesized value, then the null hypothesis cannot be rejected.

Another advantage of using this is that they help in decision-making. For instance, suppose a drug company wants to know the average weight loss achieved by a new drug. The company might conduct a study and calculate a confidence interval for the mean weight loss. Based on the confidence interval, the company can decide whether to market the drug or not.

This can also be used to compare two or more groups. For example, suppose we want to compare the mean salaries of two different groups of employees, such as male and female employees. We can calculate the mean difference in salaries between the two groups, and then determine if the intervals overlap or not. If the intervals do not overlap, we can conclude that there is a significant difference between the mean salaries of the two groups.

In summary,  intervals are an essential tool for statisticians, researchers, and decision-makers. They provide a measure of the uncertainty associated with sample estimates and help in making informed decisions based on data.