7. Proportions Flashcards
Proportion
A fraction of individuals having a particular attribute
Binomial distribution
Probability of a given number of “successes” from a fixed number of independent trials
Binomial formula
Gives the probability of X successes in n trials
where n = number of trials, p = probability of success generally according to null hypothesis (ex. 0.5)
Pr[X] ={ (n/X)*p^X } * { (1-p)^(n-X) }
where (n/X) = n! / X!(n - X)!
Meaning of ! in equations
FACTORIAL!
product of multiplying given number and every number smaller than itself
ex. n! = n * n-1 * n-2 * n-3 * … 3 * 2 * 1
Binomial Distribution
provides the probability distribution for the number of “successes” in a fixed number of independent trials, when the probability of success is that same in each trial
assumptions of the binomial formula
the number of trials (n) is fixed
separate trials are independent
the probability of success (p) is the same in every trial
meaning of (n /X) in binomial equation
“n choose X”, number of unique ordered sequences of successes and failures that yield exactly X successes in n trials
what are (n/0) and (n/n) equal to in binomial formula
1!!! because 0! = 1
What two things does the binomial distribution describe
the sampling distribution for the NUMBER of successes in a random sample of n trials, but also the PROPORTION of successes
u = np ???????????
sigma^2 = np ( 1 - p ) ????????????c
proportion of successes in a sample equation
p hat = X / n
(number of “successes” over total sample size)
p hat therefore ESTIMATE of p
mean of sample proportions
= p
variance of sample proportions
p(1-p) / n
Standard error of the estimate of a proportion
SE[phat] = sqrt{ [phat (1 - phat)]/n }
Size of standard error for larger sample size
lower standard error!!!
To determine the 95% confidence interval / Agresti-Coull confidence interval for an estimate:
- determine p’
p’ = (X + 2) / (n + 4)
- plug p’ into equation:
p’ - 1.96 * (sqrt { [p’ * (1 - p’)] / [n + 4]) </= p </= p’ + 1.96 * (sqrt { [p’ * (1 - p’)]
Binomial test
Uses data to test whether a population proportion p matches a null expectation for the proportion
H0 for binomial test
The relative frequency of successes in the population is p0
HA for binomial test
The relative frequency of successes in the population is NOT p0
How to use binomial equation in calculating P-value
P-value (P) = 2 * (Pr[X] + Pr[X-1] + Pr[…] + Pr[1])
