#### RESULTS

## What is confidence interval?

A simple definition of the confidence interval is a range of values that has the inclusion of a population parameter. The value of this parameter is unknown. When it comes to the best calculation option, using a confidence interval calculator is the finest alternative.

## Confidence Interval Formula

The confidence interval can be calculated by using the following formula:

`\(\textbf{Lower bound Value = Mean Value (x) - Margin of Error}\)`

`\(\textbf{Upper Bound Value = Mean Value (x) + Margin of Error}\)`

Confidence interval depends on the standard error and margin of error. The formula for standard deviation can be expressed as:

Standard error `\(= \dfrac{\sigma}{\sqrt{n}}\)`

The formula for the margin of error can be written as:

Margin of error `\(= \text{standard error} \times \text{Z}(0.95)\)`.

Where Z (0.95) represents the z-score equal of 95% confidence level. If you use a particular confidence level, the correct z-score must be determined instead of that factor.

## How to calculate Confidence interval?

Here we will illustrate the method to find a confidence interval using the above formulas. Follow these steps to calculate the confidence interval:

- Identify and write down the values
- Calculate the standard error using standard error formula
- Calculate the margin of error using the margin of error formula
- To calculate the upper and lower bound of the confidence interval, add and subtract the margin of error from the mean value.

### Example:

How to Find a 95% confidence interval?

Let's understand the procedure of confidence interval calculation by using an example.

Suppose that there is a sample of **50 bowls** with different sizes. The standard deviation is **4**, and the mean size is 10. What will be the confidence interval?

**Solution:**

We will calculate the confidence interval by using the above formulas step by step. Follow the below steps to get a confidence interval for the given values:

** Step 1: **Identify and write down the values.

`\(\sigma= 4, n = 50, \Mu=10\)`

** Step 2: **Calculate the standard error using the standard error equation.

**Standard Error** `\(= \dfrac{\sigma}{\sqrt{n}} = \dfrac{4}{\sqrt{50}} = 0.56\)`

** Step 3: **Calculate the margin of error using the margin of error equation. The Margin of Error would be determined on the basis of the standard error value calculated above.

**Margin of Error** `\(= \text{Standard error} \times Z (0.95)\)`

Here the term `\(Z (0.95)\)` defines the value of Z score when the confidence interval is 95%. Refer to this Z table to get the Z value. In this case, `\(Z =1.758\)`.

**Margin of Error** `\(= 0.56 \times 1.758 = 0.98\)`

** Step 4: **Now, to calculate the upper and lower bound of the confidence interval, add and subtract the margin of error from the mean value.

**Mean** = 10

Hence, the range will be written as:

`\(10 - 0.98 \leftrightarrow 10 + 0.98\)`

So,

**Lower bound** `\(= 9.02\)`

**Upper bound** `\(= 10.98\)`

## How to Use our Confidence Interval Calculator?

To use our confidence interval calculator:

- Select a value from raw data or Mean and SD.
- Select a confidence level from the list.
**95**confidence level will be selected by default if you don't choose a confidence level. - Enter the mean value and standard deviation value in the given input boxes.
- Enter the sample size in the given input box.
- Press the
**Calculate**button to see the output.

The confidence interval calculator will instantly calculate the confidence interval with the selected confidence level and show you the confidence interval as well as the margin of error. You can use our standard deviation calculator to calculate the standard deviation for the confidence interval.

## FAQ's

### What does a 95% confidence interval mean?

A 95% confidence level implies that 95% of the time, the findings will represent the outcomes from the entire population if the study or experiment was replicated. Occasionally, because of time or costs, you can't interview everybody.

### What is a good confidence interval?

Your statistical accuracy depends on the variability and sample size. A low variability or larger sample size corresponds to a narrower confidence interval with a lower margin of error. A higher variability or smaller sample size can lead to a larger interval of confidence with a greater margin of error. The confidence level also affects the interval range. The interval will not be as tight if you want a higher confidence level. A 95% or higher confidence interval is optimal.

### What is a confidence interval in simple terms?

A simple definition of the confidence interval is a range of values that has the inclusion of a population parameter. The value of this parameter is unknown. When it comes to the best calculation option, using a confidence interval calculator is the finest alternative.

### Why is confidence interval important?

Confidence intervals take into account the sample size and the possible population variance and give us an estimate of the real response. The confidence interval is a warning sign that you can use a grain of salt to take this sample result because you cannot be accurate than this answer.

### What is a confidence interval in statistics?

The Confidence Interval in statistics is a form of estimation based on the statistics of the data observed. This suggests a number of credible values for an unidentified parameter. The interval has the accompanying level of confidence that the true parameter is within the range suggested.

### Is a smaller confidence interval better?

A tighter confidence interval seems to indicate a smaller chance of an occurrence of observation in this interval since our precision is higher. A 95 percent confidence interval is also tighter than a broader 99 percent confidence interval. The 99% confidence interval is reliable than 95% confidence interval. So no, the smaller confidence interval is not better.

### Which is better 95 or 99 confidence interval?

The 99% confidence interval is precise than the 95% confidence interval.

### Where would you use a confidence interval in everyday life?

Whenever we estimate or forecast a number, we usually include a confidence interval to show just how wrong we think we might be. In many cases, this is a pretty good way of representing the error inherent in our data and our method of forecasting.

Moreover, it is used anywhere statistics are used. At least, where statistics are used as a method of estimation or prediction. So, that includes:

- All the sciences (soft, hard, behavioral, medical)
- Business
- Investing
- Politics
- Advertising

## References:

- Confidence intervals | Statistics and probability | Math | Khan Academy
- Mcleod, S. | What are confidence intervals? | examples | source-Simply Psychology
- Confidence Intervals. (1997)-How to find confidence interval | from the source of stat.yale.edu/
- Statements of probability and confidence intervals | bmj.com