# Is it Acceptable to Sum Multiple Observations for Proportion in Inferential Statistics and Hypothesis Testing?

Suppose I'm sampling two groups for an experimental research where I'm measuring how people respond to a certain type of question under the interference of certain factors in one of the groups, and without the interference in another.

Aiming to have a 0.95 confidence within a 0.05 interval, the sample size is almost a prohibitive target to seek twice (one time for each group) due to certain project constraints.

So suppose we have i.e one variable x1 to measure. Rather than measuring it once for each participant (by asking one question), what if we measure it twice for each participant through asking him/her two equivalent questions instead (i.e. same difficult level, structure, etc, but different numbers)?

Example: Suppose we ask two questions q1 and q2 to measure x1 variable, and got the following: - Participant 1 answered q1 wrong and q2 right. - Participant 2 answered both q1 and q2 right. - Participant 3 answered q1 right and q2 wrong.

So I attribute the following to x1: n=6; 4 right answers, p=0.67

Would it be a valid procedure for both drawing an inference for x1? What if we, later, compare the measurements of x1 in the two groups for a Z test?

Is there any literature discussing such approach?

Also, is there anything to do to check whether the questions were really equivalent (i.e. like a MacNemar test)? Would it be necessary?