Question

How do you construct a 95% confidence interval?

Answer 1

Hint: We first express the concept of Confidence levels. We take an arbitrary example to understand the concept better.

Complete step by step solution:
Confidence levels are expressed as a percentage (for example, a 95% confidence level). It means that should you repeat an experiment or survey over and over again, 95 percent of the time your results will match the results you get from a population (in other words, your statistics would be sound). Confidence intervals are your results and they are usually numbers.
We constructed a 95% confidence interval for an experiment that found the sample mean temperature for a certain city in August was $\bar{x}=100$, with a population standard deviation of $\sigma =17.50$. There were 40 samples in this experiment.
The $\alpha$-level is ${\displaystyle \frac{1-.95}{2}}=.025$.
The formula is $\bar{x}\pm (z\times {\displaystyle \frac{\sigma }{\sqrt{n}}})$. Here n is the number of samples. z is from the standard distribution tables (in the reference), and is $1.96$for a CI of 95%.
So, $\bar{x}\pm (z\times {\displaystyle \frac{\sigma }{\sqrt{n}}})=100\pm (1.96\times {\displaystyle \frac{17.5}{\sqrt{40}}})=100\pm 5.42=[94.58,105.42]$.

Note:
For example, if we survey a group of pet owners to see how many cans of dog food, they purchase a year. We test our statistics at the 99 percent confidence level and get a confidence interval of (200, 300). That means you think they buy between 200 and 300 cans a year. We are super confident (99% is a very high level) that our results are sound, statistically.