Question

COIN TOSSES In a large class of introductory Statistics students, the professor has each student toss a coin 16 times and calculate the proportion of his or her tosses that were heads. The students then report their results, and the professor plots a histogram of these several proportions.

1. What shape would you expect this histogram to be? Why?

2. Where do you expect the histogram to be centred?

3. How much variability would you expect among these proportions?

4. Explain why a Normal model should not be used here.

The answer is

1, symmetric

2, 0.5

3, 0.125

4. np=8<10

Please explain why the third question has answer 0.125. how did you get it?

Answer 1

.125 is given as the answer because here standard deviation of sample proportion has been used as a measure of variability. Let me explain it properly.

Here the coin is assumed to be an unbiased coin. So probability of getting head in one toss is 1/2 (=.5). Now the coin has been tossed 16 times by each student. Let's define a random variable ''X" where X is the total number of heads obtained in 16 tosses by a student.

So proportion of heads in 16 tosses is X/16.

Let us define "p" as p=X/16.

Now each student reports value of p as observed by him/her.

Now the professor plots histogram with independent sample observations on "p"(independent as students toss independently).

So expected variability among this proportions is standard deviation of "p".

Now we know that X follows binomial(16, .5) distribution. So variance of X= Var(X)= 16*.5*.5 =4 [using, variance of binomial(n,p) distribution is n*p*(1-p) ]

Now Var(p) =

Now standard deviation of p= = =.125

So required answer is .125