This question aims to develop key concepts regarding the **confidence intervals** and the **sample means** which are the fundamental concepts when it comes to the application of **statistics in practice**, especially in **data science** and** project management**, etc.

By definition, a **confidence interval** is basically a **range of values.** This range is **centered on the mean value** of the given sample. The **lower limit** of this range is calculated by **subtracting the variance from the mean value**.

\[ \text{ lower limit } \ = \ \bar{ x } \ – \ \sigma \]

Where $ \bar{ x } $ is the **sample mean** and $ \sigma $ is the** variance** value for the given sample. Similarly, the **upper limit** is obtained by **adding the variance to the mean** value.

\[ \text{ upper limit } \ = \ \bar{ x } \ + \ \sigma \]

The physical **significance** of this confidence interval depicts that all the **values that you expect** from a certain population **will fall within range** with some confidence percentage.

For example, if we say that the **95% confidence interval** of employee attendance of a company is ( 85%, 93% ), then it means that** we are 95% confident** that the **employee attendance will fall between 85% to 93%** range, where the mean value is 89%.

One might say that confidence intervals are a **way of describing probabilities in statistics**. Mathematically, the confidence interval can be calculated by using the following formula:

\[ CI \ = \ \bar{ x } \ \pm \ z \ \dfrac{ s }{ n } \]

where $ CI $ is the **confidence interval**, $ \bar{ x } $ is the **sample mean**, $ s $ is the sample** standard deviation**, $ z $ is the **confidence level** value and $ n $ is the **sample size**.

Given a confidence interval, the **sample mean can be calculated** using the following formula:

\[ \bar{ x } \ = \ \dfrac{ \text{ lower limit } \ + \ \text{ upper limit } }{ 2 } \]

## Expert Answer

**Given the interval (114.4, 115.6):**

\[ \text{ lower limit } \ = \ 114.4 \]

\[ \text{ upper limit } \ = \ 115.6 \]

**The sample mean can be calculated using the following formula:**

\[ \bar{ x } \ = \ \dfrac{ \text{ lower limit } \ + \ \text{ upper limit } }{ 2 } \]

**Substituting values:**

\[ \bar{ x } \ = \ \dfrac{ 114.4 \ + \ 115.6 }{ 2 } \]

\[ \Rightarrow \bar{ x } \ = \ \dfrac{ 230 }{ 2 } \]

\[ \Rightarrow \bar{ x } \ = \ 115 \]

## Numerical Result

\[ \bar{ x } \ = \ 115 \]

## Example

Given a confidence interval (114.1, 115.9), **calculate the sample mean.**

**For the given interval:**

\[ \text{ lower limit } \ = \ 114.1 \]

\[ \text{ upper limit } \ = \ 115.9 \]

**The sample mean can be calculated using the following formula:**

\[ \bar{ x } \ = \ \dfrac{ \text{ lower limit } \ + \ \text{ upper limit } }{ 2 } \]

**Substituting values:**

\[ \bar{ x } \ = \ \dfrac{ 114.1 \ + \ 115.9 }{ 2 } \]

\[ \Rightarrow \bar{ x } \ = \ \dfrac{ 230 }{ 2 } \]

\[ \Rightarrow \bar{ x } \ = \ 115 \]