In this article, we are going to talk in depth about confidence interval including its meaning, importance, statistical significance, and so on.
Confidence intervals(CI) are basic statistical concepts. A confidence interval represents the possibility that a parameter will fall between two values around the mean. It expresses the degree of uncertainty surrounding a certain result. We normally use them with a margin of error. Confidence intervals are calculated and expressed as a percentage, mainly using confidence levels of 95 percent or 99 percent. We’ll have a better idea of those percentages in a few moments.
The confidence interval estimate (CI) for both continuous and categorical variables is a range of likely values for the population proportion based on:
- The point estimate, such as the sample mean
- The researcher’s desired level of confidence (usually 95 percent)
- The sampling variability, or the point estimate’s standard error
The following is the formula for calculating Confidence Interval for the mean:
- CI = The confidence interval
- X = The sample mean
- Z = The desired confidence level’s Z value
- s = The sample standard deviation
- n = The number of elements in a sample
If you’re still uncertain about what a confidence interval is or what it does, let me explain it to you in a different way.
After All, What Is Statistics?
The first thing that you must note is how statistics is all about estimation/prediction/forecasting. We devise various methods only to be able to predict things and draw conclusions. We also happen to find various ways to make sure that our prediction is accurate(Prediction accuracy plays a role there.) It all begins with estimation and then analyzing that estimation based on past events and examining its accuracy. Now that we’ve clarified this notion, let’s look at whether the confidence interval is a tool for estimating or not.
Better Explanation for Understanding CI
You know how, normally, when we want to draw conclusions about a population, we take a sample from that population. We ask them to fill out certain survey forms or polls. And then the results that we get from those are what we assume are the results for the entire group of population for which we want to draw conclusions. But, if you asked the entire population to fill out the survey forms and polls, would you get the same conclusion as you did from examining the sample? This is when the confidence interval comes into play. It shows how certain you are that the findings of a poll or survey are representative of what you would expect to discover if you had asked the entire population to fill out the same survey forms or polls.
Assumptions for a Confidence Interval
When attempting to calculate a confidence interval, the following assumptions must be made:
- Random Sample: Your sample must be chosen at random. Always make sure that you’re dealing with a random sample.
- Normal Condition: The sample proportions’ sampling distribution has a normal form. To make the assumption that it is fairly normal, the criterion we follow is to expect more than 10 successes and failures for each sample. For example, if your sample size was 20 but your actual proportion was 70%, you are not meeting the criterion. Always make sure that you have more than or equal to 10 successes.
- Independence Condition: The 10% rule applies to the independence criterion. For example, if we sample without a replacement, our sample size must be smaller than 10% of the population size.
What Is a 95% Confidence Interval?
A confidence level of 95% indicates that if the survey or experiment were repeated, the data would match the results from the full population 95% of the time. As I demonstrated with the confidence interval example previously, we don’t always have the time to poll/survey everyone. But what would happen if we did? With a 95% confidence level, you can almost be confident that your results are the same as if you surveyed everyone.
Finding CI for the Mean using Excel
A confidence interval for the mean is a method of determining the real population mean. Instead of providing you with a single value for the mean, a confidence interval provides both, a lower and upper estimate.
The following is the formula for calculating CI for the Mean:
In Excel, we will compute the 95 percent confidence range for the mean. The following is the sample data that we will be using:
The first step will be to enter this information into a single column in Excel. I entered the information into cells A1 through A20.
This is what the data looks like currently:
The second thing we’ll do is go to the “Data” tab and then to the “Data Analysis” option in the upper right corner.
After that, we’ll click “Descriptive Statistics” and then “OK.”
We must now fill in the blanks to convey to Excel what we need to generate.
- In the Input Range box, enter your input range. I’ve typed in A1:A20.
- In the Output Range box, enter an output range. This is where the results will be displayed. I’ve entered B1 for this.
- Select the “Summary Statistics” check box and set your preferred confidence level in the “Confidence Level for Mean” check box. I’ve entered 95 here because that’s what we’re attempting to calculate.
This is what your tab will look like:
Once you have clicked “OK“, the following are the results you will encounter:
The confidence level we get for this data is 45.82902071. A confidence level of 95% or higher is generally considered ideal.
Asymmetric Confidence Interval
The term “asymmetric confidence interval” simply refers to the fact that the point estimate does not fall directly in the middle of the CI. If you don’t follow the assumptions, this can happen. However, this frequently arises when the interval contains a random error or unsystematic error or systematic bias. Systematic bias is a type of measuring inaccuracy that is directional. Positive systematic bias increases the upper bound of a confidence interval, whereas negative systematic bias decreases the lower bound of a confidence interval.