# Confidence Interval

{% hint style="success" %}
Confidence level represents the probability that the unknown parameter lies in the stated interval. The level of confidence can be chosen by the investigator.
{% endhint %}

This proposes a range of plausible values for an unknown parameter. The interval has an associated confidence level that the true parameter is in the proposed range.

{% hint style="info" %}
Imagine you want to find the mean height of all the people in a particular US state. You could go to each person in that particular state and ask for their height, or you can do the smarter thing by taking a sample of 1000 people in the state. 
{% endhint %}

Then you can use the mean of their heights (Estimated Mean) to estimate the average heights in the state (True Mean).

### Calculating Confidence Interval

![](https://2552912007-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LzGBVuquaFNrwdmJna0%2F-LzZD60gNvNhQEILfcl3%2F-LzZErqXCdCji33fsM4M%2FScreen%20Shot%202020-01-26%20at%204.45.55%20PM.png?alt=media\&token=69c79414-2e02-40ef-aa16-4d0eeea03cd1)

We cast a net from the value we know $$\overline{x}$$ .

To get such ranges or intervals, we go `1.96 SD` away from $$\overline{x}$$ (the sample mean) in both directions. And this range is the `95%` confidence interval.

Now, when we say that, we estimate the true mean to be $$\overline{x}$$ (the sample mean) with a confidence interval of \[ $$\overline{x}-1.96\sigma, \overline{x}+1.96\sigma$$ ], we are literally saying that: It is with `95%` probability that the true population mean is within these Confidence Interval limits.

> When you take 99% CI, you essentially increase the proportion and thus cast a wider net with three standard deviations.

$$
\overline{x} \pm z \frac{s}{\sqrt{n}}
$$

Here, $$\overline{x}$$  is the sample mean (mean of the 1000 heights sample we took). $$z$$ is the no. of standard deviations away from the sample mean (1.96 for 95%, 2.576 for 99%), level of confidence we want. $$s$$ is the standard deviation in the sample. $$n$$ is the size of the sample.

### References

{% embed url="<https://towardsdatascience.com/confidence-intervals-explained-simply-for-data-scientists-8354a6e2266b>" %}