Question

Consider a binomial experiment. If the number of trials is increased, what happens to the expected value, if the number of trials is decreased, what effect would occur? With regard to the standard deviation, illustrate how and why the larger the deviation number, the more difficult it would be to predict an outcome. Use specific examples to illustrate.

Answer 1

In a binomial distribution, the expected value, E(X) is "np" and the standard deviation, Std(X) is ​​​​​​.

Where, X =number of successes in the binomial experiment; n =sample size =number of trials in the binomial experiment; p =proportion of success on a single trial; q =1 - p.

If n is increased or decreased:

Let p =0.5; n =50; so, q =1 - p =0.5

E(X) =50*0.5 =25

If n increases to 80, E(X) =80*0.5 =40

If n decreases to 30, E(X) =30*0.5 =15

Thus, E(X) increases with an increase in the number of trials and E(X) decreases with a decrease in the number of trials.

The larger the standard deviation:

"The larger the standard deviation, the more difficult it would be to predict an outcome and the smaller the standard deviation, the less difficult it would be to predict an outcome".

Let us take specific examples to illustrate this:

Consider the data, 2,2,2,2,2,2,2,2,2,2. Now predict the next outcome. Obviously, we predict it to be 2. It's not difficult because, in this case, the standard deviation is the smallest being 0.

Consider the data, 2,3,2,2,2,3,3,3,2,3.Now predict the next outcome. We predict it to be either 2 or 3. We are not sure of a single outcome here because here the standard deviation has increased to 0.50. So, it is more difficult to predict compared to the previous data where standard deviation is less.

Now, consider the data, 2,4,6,6,3,5,3,9,12,10. Now predict the next outcome. It is even more difficult to predict compared to previous data sets because now, the standard deviation has increased to 3.1623

(The formula which was used here to calculate the standard deviation is: where, N =Number of observations; =mean of the data; =Observed value).

Thus, the larger the deviation number, the more difficult it would be to predict an outcome.