Question

GE 5.52 Find the mean and the standard deviation. If we toss a fair coin 150 times, the number of heads is a random variable that is binomial. (a) Find the mean and the standard deviation of the sample propor- tion of heads. (b) Is your answer to part (a) the same as the mean and the standard deviation of the sample count of heads in 150 throws? Explain your answer

Answer 1

The experimernt is tossing a fair coin.

Let X be the random variable that number of heads.

Here X ~ Binomial (n = 150, p = 1/2)

The pmf of X =x is,

$P(X = x) = (150 C x) * (\frac{1}{2})^x * (1-\frac{1}{2})^(150-x)$ x = 0,1,2,.....................150

a) Mean and standard deviation of the sample proportion :

We know that,

Mean = p = 1/2 = 0.5

$standard deviation (sd) = \sqrt{\frac{p*(1-p)}{n}}$

  $standard deviation (sd) = \sqrt{\frac{\frac{1}{2}*(1-\frac{1}{2})}{150}}$

sd = 0.041

b) Let X be the random variable that number of heads.

sample size (n) = 150

sample proportion (p) = 1/2

Mean = 150*1/2 = 75

Standard deviation = $\sqrt{n*p*(1-p)}=$ sqrt (150*1/2 * (1-1/2)) = 6.12

No both the values are not same.

Because in first part we take random variable as sample proportion while in second case we take random variable as number of heads.

Both the distributions are same but values are different.