Question

We are interested in the proportion of students in our data analysis class who love the normal distribution. Our class has about 30 students. Suppose that the true proportion in our class who love the normal distribution is 20%. (A surprisingly low value. Ha ha.) We sample 10 students out of 30 to be surveyed about their love for the normal distribution. Let X= # in the sample of 10 who love the normal distribution. Explain why we cannot say the X has a B(n=10,p=0.2) distribution.

Answer 1

Since there are only 30 students in the class and only 20% love the normal distribution which means only 0.20*30 = 6 students in the class of 30 love the normal distribution.

Now, when we sample 10 students out of these 30, the maximum no. of students possible in the sample who love normal distribution is 6, i.e. the maximum possible value of X is 6 and the probability that there are more than six students in the sample who love normal distribution is exactly equal to zero, i.e., P(X>6) = 0

But if we use Binomial(n=10,p=0.2) distribution to describe the distribution of X, P(X>6) 0, which is a contradiction.

Thus, we cannot say that X has a B(n=10,p=0.2) distribution.

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