http://www.ck12.org Chapter 7. Sampling Distributions and Estimations
Aspincreases to 0.5, the skewness disappeared and the distributions achieved perfect symmetry. The symmetrical,
mound-shaped distribution remained the same for all values ofn.
n=5 and p= 0. 75 n=10 and p= 0. 75 n=20 and p= 0. 75For the larger value ofp, the binomial distributions are skewed toward the lower values ofx. Asnincreases, the
skewness decreases and the distributions gradually move toward being more normal.
BecauseE(X) =n p=μx, the value increases with bothnandp. Asnincreases, so does the standard deviation but
for a fixed value ofn, the standard deviation is largest aroundp= 0 .5 and reduces aspapproaches 0 or 1.
Example:Suppose you flip a fair coin 12 times.
a) What is the probability that you will get exactly 5 heads?
b) Exactly 25% heads?
c) At least 10 heads?
Solution:LetXrepresent the number of heads from 12 flips of the coin. Exactly 5 heads:
a)
(
n
k)
=
n!
k!(n−k)!P(X= 5 ) = 792 ( 0. 5 )^12 = 0. 1933
(
12
5
)
=
12!
5!( 12 − 5 )!
(
12
5
)
= 792
b) 25% of 12 heads means getting 3 heads.