Inferential Statistics

Objective — Find the mean of the population from some samples, within a confidence level and confidence interval.

Steps —
1. Find the mean of one sample : xbar
2. Find the number of elements in the sample : n
3. Find the standard deviation of the sample : s
4. Find the standard error of the means of the sampling distribution of all the samples taken : SEM (This is the standard deviation of the sampling distribution of the means of all the samples taken. As per CLT it is normally distributed)
SEM = mean of the population (meu)/ square root of n
(xbar is is an approximation of meu as per CLT)
5. Given the confidence interval, finding the confidence level :
Say the interval is +-1.98. Then this 1.98 is directly the z-score, and find the probability of this z score for both positive and negative one. Then subtract the probability score of the negative from the positive one. This gives you the confidence level.
6. Given the confidence level, finding the confidence interval :
Find the zscore corresponding to the confidence interval given as per the following table :

Fig 1 : Confidence level to z score map

Find Margin of Error (MOE) : Now, you have the z score of the sampling distribution. If you multiply it with the standard deviation of the sampling distribution which is SEM (Standard Error of the Means) then you will get the +- interval values.
Confidence interval is the xbar +- MOE!