Statistics Summary Calculator

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Discrete Distrubutions

Distribution	Formula	Mean μ	Variance σ2	Moment Generating Function
Bernoulli Distribution	p if k = 1, 1 - p if k = 0, else 0	p	p(1 - p)	1 - p + pet
Binomial Distribution	f(k;n,p)  =   n! * pk(1 - p)n - k    k!(n - k)!	np	np(1 - p)	(1 - p + pet)n
Negative Binomial Distribution	f(n;k,p)  =   (n - 1)! * pk(1 - p)n - k    (k - 1)!(n - k)!		k(1 - p) p2	(1 - p)r (1 - pet)r
Geometric Distribution	P(x = n) = p * (1 - p)(n - 1)			pet 1 - (1 - p)et
Poisson Distribution	P(k; λ)  =   λk    eλk!	λ	λ	eλ(et - 1)
Hypergeometric Distribution	P(x;n,N,k)  =   (kCx) * (N - kCn - x)    NCn		nk(N - k)(N - n) N2(N - 1)	N/A
Multinomial Distribution	ƒ(x0!·x1!·x2...&middotn;x;n,θ0,θ1,θ2,...,θn)  =   n!(θ0 · θ1 · θ2 · ... · θn)    x0 · x1 · x2 · ... · xn	npi	npi(1 - pi)	(Σpieti)n
Uniform Distribution	P(k; λ)  =   λk    eλk!	½a + b	(b - a)2 12	etb - eta t(b - a)

Test Statistic Decision Tree:

Type	Keywords	Sample Size	Use Test
Z-score	Mean, Average	Greater than 30 or population σ is known	z  =   X - μ    σ/√n
t-score	Mean, Average	30 or less: population σ is not known	t  =   x - μ    s/√n
proportion-score	Proportion (Test p), Fraction, Percentage, Rate, Probability	more than 30	z  =   p^ - p0    √p0q0/n
Variance σ2	Variance, Variability, Spread	N/A	χ2  =   (n - 1)s2    σ2
Equal Variances σ12 = σ22	Equal Variances, Ratio or Difference in Variances	N/A	F  =   σ12    σ22

Confidence Interval Decision Tree

Type	Keywords	Sample Size	Use Test
Test the Mean	Confidence Interval, Mean, Average	Greater than 30	X - zscoreα/2 * s/√n < μ < X + zscoreα/2 * s/√n
Test the Mean	Confidence Interval, Mean, Average	30 or less	X - tscoreα * s/√n < μ < X + tscoreα * s/√n
Test the Variance	Confidence Interval, Variance	Greater than 30	(n - 1)s2/χ2α/2 < σ2 < (n - 1)s2/χ21 - α/2
Test the Standard Deviation	Confidence Interval, Standard Deviation	Greater than 30	Square Root((n - 1)s2/χ2α/2) < σ2 < Square Root((n - 1)s2/χ21 - α/2)
Test the Proportion	Confidence Interval, Proportion, percentage, rate, Population	Greater than 30	(n - 1)s2/χ2α/2 < σ2 < (n - 1)s2/χ21 - α/2
Test the Difference of Means	Confidence Interval, Difference of Means	Greater than 30	(x1 - x2) - zscoreα x √a < μ1 - μ2 < (x1 - x2) - zscoreα x √a
Test the Difference of Means	Confidence Interval, Difference of Means	30 or less	(x1 - x2) - tscoreα x √a < μ1 - μ2 < (x1 - x2) - tscoreα x √a
p^ Confidence Interval	Confidence Interval (test p), criteria,characteristic, proportion	30 or less	p^ - zα/2σ√p(1 - p)/n < p < p^ + zα/2√p(1 - p)/n

Sample Size Decision Tree:

Type	Keywords	Use Test
Sample Size for μ	Sample Size, average, mean	n  =   Z-scoreα/22 x σ2    SE2
Proportion Sample Size	Sample Size, Proportion, Population, Percentage, Rate	n  =   Z-score2 x p x (1 - p)    SE2
μ1 - μ2 Sample Size	Sample Size, Difference of Means, μ1 - μ2	n  =   Z-score2(σ12 + σ22)    ME2
p1 - p2 Sample Size	Sample Size, Difference of p, p1 - p2	n  =   Z-score2(p1q1 + p2q2)    ME2

Hypothesis Testing Decision Tree

p-value Significance Test (observed level of significance):

Find your z-score, then find the probability in the z-table associated with that score, and if α > probability (p-value), reject H₀

Hypothesis Testing Errors:

Type I error - Reject null hypothesis H₀ when H₀ is TRUE: Probability = α
Type II error - Accept null hypothesis H₀ when H₀ is FALSE: Probability = β
Power of the Test = Probability you Reject null hypothesis H₀ when H₀ is FALSE: --> 1 - β
Note: It is a bigger mistake to make a Type II error than a Type I error

Finite Population Correction Factor:

If n/N > 0.05, then you multiply your confidence interval by the following factor

√N - n

√N

Cov(X,Y)  =	Σ(Xi - X)(Yi - Y)
	n

Correlation Coefficient (r)  =	Cov(X,Y)
	sxsy

β  =	Σ(Xi - X)(Yi - Y)
	Σ(Xi - X)2

Least Squares Regression Line ← α = Y - βX
y^ = α + βx where α is the y-intercept for the line that contains the points in X & Y and β is the is the slope of the line that the set of points lies on.
α & β are designed such that they produce the smallest possible SSE defined below
Sum of Squares about the Mean (SSM) = (y_i - y)²
Square of the Residual Difference (SSE) (y_i - y^_i)²
SSE represents the difference between the straight line that we create and the plotted points from our data

Coefficient of Determination (r2)  =	SSM - SSE
	SSM

Large Sample Condition Requirement:

1. A random sample is selected from the target population.
2. The sample size n is large (i.e., n ≥ 30). (Due to the Central Limit Theorem, this condition guarantees that the test statistic will be approximately normal regardless of the shape of the underlying probability distribution of the population.)