Statistical Inference

What is Inference?

Important

Statistical inference is the process of drawing conclusions about the entire population based on the information in a sample.

In this Section we will examine samples from populations and find procedures for estimating parameters such as means and sd. We will also devise procedures for comparing means and variances across more than one population. The conditions that make these procedures possible and accurate will also be studied and we will find alternative methods when those assumptions breakdown.

Based on our ideas of data and types of variables, here is a table of what we may infer, based on the underlying data:

Data Types and Inference

Variable(s)	Estimating What?	Population Parameter	Sample Statistic
Single Qual variable	Proportion	p	\(\hat{p}\)
Single Quant variable	Mean	\(\mu\)	\(\bar{x}\)
Two Qual Variables	Difference in Proportions	\(p_1 -p_2\)	\(\hat{p_1} - \hat{p_2}\)
One Qual, one Quant	Difference in Means	\(\mu_1 - \mu_2\)	\(\bar{x_1}-\bar{x_2}\)
Two Quant variables	Correlation	\(\rho\)	r

We will examine inference procedures for all these cases.

An Idea to Encourage You: Stats Lessons from Sholay!!

Gabbar: “Kitne Aadmi thay?
Stats Teacher: How many observations do you have? n < 30 is a joke.

Gabbar: Kya Samajh kar aaye thay? Gabbar khus hoga? Sabaasi dega kya?
Stats Teacher: What are the levels in your Factors? Are they binary? Don’t do ANOVA just yet!

Gabbar: (Fires off three rounds ) Haan, ab theek hai!
Stats Teacher: Yes, now the dataset is balanced wrt the factor (Treatment and Control).

Gabbar: Is pistol mein teen zindagi aur teen maut bandh hai. Dekhte hain kisko kya milega.
Stats Teacher: This is our Research Question, for which we will Design an Experiment.

Gabbar: (Twirls the chambers of his revolver) “Hume kuchh nahi pataa!”
Stats Teacher: Let us perform a non-parametric Permutation Test for this Factor!

Gabbar: “Kamaal ho gaya!”
Stats Teacher: Fantastic! Our p-value is so small that we can reject the NULL Hypothesis!!

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