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
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