Consider making an inference about $p_1-p_2$, where there are $x_1$successes in$n_1$binomial trials and $x_2$successes in$n_2$binomial trials.

a. Describe the distributions of $x_1$and $x_2$.

b. Explain why the Central Limit Theorem is important in finding an approximate distribution for$\left({\hat{p}}_1-{\hat{p}}_2\right)$

We have, for the first trial

The number of binomial trials is $n_1$and $x_1$there are successes

For the second trial

There are binomial trials $n_2$, and there are$x_2$ successes.

When we have a prefixed number of trials to get the desired number of successes with a fixed probability of success throughout the experiment, the process will be well executed by binomial probabilities.

The central limit theorem helps to find the approximate distribution for the sample mean when the sample size is large enough (at least 30) regardless of the original distribution.

Hereand can be viewed as means of the number of successes per trial in the respective samples.

Hence the distribution of can be approximated using the central limit theorem, as it represents the difference in the sample mean for the number of successes per trial.