Formula_sheet

ST104a Statistics 1
Examination Formula Sheet
Expected value of a discrete random variable: Standard deviation of a discrete random variable: N

µ = E[X] =

√

pi xi

σ=

N

σ2 =

pi (xi − µ)2

i=1

i=1

Finding Z for the sampling distribution of the sample mean:

The transformation formula:
Z=

X −µ σ Z=

Finding Z for the sampling distribution of the sample proportion:
Z=

Confidence interval endpoints for a single mean (σ known): σ x
¯ ± z√ n P −π π(1−π) n

Confidence interval endpoints for a single mean (σ unknown):

Confidence interval endpoints for a single proportion:

s x ¯ ± tn−1 √ n p±z

Sample size determination for a mean: n≥ ¯ −µ
X
√ σ/ n

Sample size proportion: Z 2σ2 e2 determination

n≥

Z-test of hypothesis for a single mean (σ known): ¯ −µ
X
√
Z=
σ/ n

p(1 − p) n for

a

Z 2 p(1 − p) e2 t-test of hypothesis for a single mean (σ unknown): t=

1

¯ −µ
X
√
S/ n

Z-test of hypothesis proportion: for

a

single

Z-test for the difference between two means
(variances known):

p−π

Z∼
=

Z=

π(1−π) n t-test for the difference between two means
(variances unknown): t= 1 n1 +

+

(¯ x1 − x
¯2 ) ± tn1 +n2 −2

1 n2 Pooled variance estimator:
Sp2 =

σ12 n1 t=

sd x ¯d ± tn−1 √ n ¯ d − µd
X
√
Sd / n

Z=

(P1 − P2 ) − (π1 − π2 )
P (1 − P )

R1 + R 2 n1 + n2

(p1 − p2 ) ± z

χ2 test of association:

i=1 j=1

1 n1 +

1 n2 Confidence interval endpoints for the difference between two proportions:

Pooled proportion estimator:

c

1
1
+ n1 n2

Z-test for the difference between two proportions: Confidence interval endpoints for the difference in means in paired samples:

r

s2p

t-test for the difference in means in paired samples:

(n1 − 1)S12 + (n2 − 1)S22 n1 + n2 − 2

P =

σ22 n2 Confidence interval endpoints for the difference between two means:

¯1 − X
¯ 2 ) − (µ1 − µ2 )
(X
Sp2

¯1 − X
¯ 2 ) − (µ1 − µ2 )
(X

p1 (1 − p1 ) p2 (1 − p2 )
+
n1 n2 Sample correlation coefficient:

(Oij − Eij )2
Eij

r=

Spearman rank correlation:

n i=1 xi yi n 2