# Scope of Inference in Statistics

Hello All,

I am doing my statistics course and I was finding a hard time writing the scope of inference reading the questions. This diagram helped me understand better what to write when we have a random sample and a random assignment.

First we have to understand what is random sampling and random assignment

Random Sampling – In this technique, each member of the population has an equal chance of being selected as subject. The entire process of sampling is done in a single step with each subject selected independently of the other members of the population

Random Assignment/Experiment –  This is nothing but random placement or assignment of subjects(human participants or animal subjects )to different groups in an experiment using randomization, such as by a chance procedure (e.g., flipping a coin) or a random number generator.

Lets take an example to illustrate this

1. Suppose we have developed a new fertilizer that is supposed to help corn yields. This fertilizer is so potent that a small vial of it sprayed over an entire field is a sufficient dose.  We find that the new fertilizer results in an average yield of 60 more bushels over the old fertilizer with a p-value of 0.0001.  Write up a scope of inference under the following study designs that generated this data.

a. We offer the new fertilizer at a discount to customers who have purchased the old fertilizer along with a survey for them to fill out. Some farmers send in a survey after the growing season, reporting their crop yield. From our records, we know which of these farmers used the new fertilizer and which used the old one.

Okay from this question we have to understand if its a random assignment and if its a random sample

Is it a Random Sample ?

No

Here the farmers self-select into the study and hence this experiment cannot be generalized to the population and results are confined only to this group.

Is this Randomly assignment ?

No

Here we do  not randomize the assignment of ‘old’ and ‘new’ fertilizer and hence no casual inference can be made.Also, we do not get to randomize the assignment to “old” and “new” fertilizer, so no causal inference can be made.(Look at the diagram)

#### Lets take another example

2.When a customer makes an order, we randomly send them either the old or new fertilizer. At the end of the season, we sub-select from the fertilizer orders and send a team out to count those farmers’ crop yields.

Is this a random sample ?

Yes

The customers where send randomly old or new fertilizers. These results can be generalized to the population(All farmers).

Is this a random assignment ?

Yes

Since we subselect from the orders we are randomly assigning the fertilizer groups to the farmers. Hence casual inference can be made .(Look at the diagram)

Hope the diagram will make some sense when you write your scope of inference.

Cheers !!

Jethin Abraham