Binominal Expansion Flashcards
How to spot a binominal question
Watch for a % (P) and an n. note a % can be a decimal or a fraction! Once you spot it list your p, n and X
ex: Flipping a coin 5 times and counting how many heads:
P = 1/2 = 0.5 b/c fair coin
n =5
x = 0,1,2,3,4,5 = possible number of heads tossed
Ux
mean of X = average number x will be = n x P
variance (x)
SigmaX^2 = np(1-p) and so SD would equal the root of that
pHat
sample proportion aka sample percentage = X/n. Ex if 7 of our sample of 20 have covid, pHat = 7/20
UpHat
the true precentage ; Uphat = p
Variance (pHat)
P(1-P) / n
Two cases of binominal probability porblems
- When n is small (n<20)
- When n is huge (n>1000)
When n is Small (4)
- List your n =
- list your p =
- List your x = 0,1,2,3,4……n. And box the values you want (K)
- solve with P(x=k) = (n choose K) p^k (1-p)^n-k ,
n choose k
denoted as (n over k with no line) = n!/K!(n-k)! recall 4! = 4x3x2x1. ex how many times can you draw 6 cards from a deck of 10? 10!/6!(10-6)!
When n is Huge (8 steps)
- list your n =
- list your p =
- list or compute pHat with pHat = x/n
- draw a bell curve centered at P
- mark pHat on curve
- Shade area
- use the z formula for phat on formula sheet to get test stat,
- use test stat to get probability
Two population hypothesis test
Same as with one sample but different test stat formula:
z= (pHat1 - pHat2)/SE(phat1-pHat2)
Confidence interval for two population
(pHat1-pHat2) +- ZStar x SE(phat1-phat2)
Which SE formula to use?
If you are testing a hypothesis with two populations, your H0 is p1 = p2 and because we always assume h0 is correct, you use the formula where p1=p2 (on sheet). For confidence intervals, we are assuming there is a difference so you use the one where p1 does not = p2 (also on sheet)
H0 for hypothesis test
p1=p2 ALWAYS
