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:
\[ \bar{ x } \ = \ \dfrac{ 114.1 \ + \ 115.9 }{ 2 } \]
\[ \Rightarrow \bar{ x } \ = \ \dfrac{ 230 }{ 2 } \]
\[ \Rightarrow \bar{ x } \ = \ 115 \]