confidence-interval.md

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

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" %}