Question

Start StatCrunch and make the following sequence selection: Applets -> Distribution demos. Next select "Binomial" and click "Compute!". In the resulting popup window experiment by using the sliders to assign approximately 0.5 to p and successively assign the values 20, 30 and 40 to n. Discuss what you see in the subsequently drawn Binomial Distribution defined by your specified values for n and p. What value on the x axis (horizontal axis) does the top of the hump of the curve correspond to. Next set p and n to their extreme values? Discuss what you observed using the fact that the x-axis represents the number of successes and the height of the vertical lines represent the probability of getting x number of successes.

Answer 1

Let p =0.5 and n =20

Let p =0.5; n =30

Let p =0.5 and n =40

In the subsequently drawn Binomial Distributions, as n is increased, the shape of the distribution became more bell shaped (more clustered around the center). However, since p =0.5, all the distributions are normal (symmetric).

The value on the x axis (horizontal axis) that the top of the hump of the curve corresponds to is the expected value of X.

X-value on the top of the hump of curve =E(X) =Expected number of successes.

Now,

Let us take extreme values for p, that is, we take p =0.1 and p =0.9 and then extreme values for n being 5 and 1000.

When p =0.1, the distribution is right skewed at n =5 and 1000.

(so, when the proportion of success on single trial is very less, we have less number of successes and at the same time, as n increased, the total probability of 1 has been shared by more number of X-values and so, the highest probability is less than that of when n is less).

When p =0.9, the distribution is left skewed at n =5 and 1000.

(so, when the proportion of success on single trial is very high, we have more number of successes and at the same time, as n increased, the total probability of 1 has been shared by more number of X-values and so, the highest probability is less than that of when n is less).

This is for extreme values of n only. If n is not extreme, then the distribution is not so much skewed as above. It will be less skewed and approximately normal even though we take extreme values for p.

For example, let n =100, then we have the following approximately normal curves at p =0.1 and 0.9.

Slightly right skewed (approximately normal).

Slightly left skewed (approximately normal).