Imperial College London Statistics Questions You are told that a 99% confidence interval for a single population proportion is (0.3676, 0.5324). i. What was

Imperial College London Statistics Questions You are told that a 99% confidence interval for a single population proportion is (0.3676, 0.5324). i. What was the sample proportion that lead to this confidence interval? ii. What was the size of the sample used? lOMoARcPSD|5093418
ST104a Statistics 1
Examination Formula Sheet
Expected value of a discrete random
variable:
µ = E(X) =
N
X
pi x i
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=p
Standard deviation of a discrete random
variable:
v
uN
uX
√
2
σ= σ =t
pi (xi − µ)2
Confidence interval endpoints for a
single mean (σ known):
σ
x̄ ± zα/2 × √
n
P −π
π(1 − π)/n
Confidence interval endpoints for a
single mean (σ unknown):
s
x̄ ± tα/2, n−1 × √
n
Sample size determination for a mean:
(zα/2 )2 σ 2
n≥
e2
Confidence interval endpoints for a
single proportion:
r
p (1 − p)
p ± zα/2 ×
n
Sample size
proportion:
determination
for
a
(zα/2 )2 p (1 − p)
e2
n≥
z test of hypothesis for a single mean (σ
known):
X̄ − µ0
√
Z=
σ/ n
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X̄ − µ
√
σ/ n
t test of hypothesis for a single mean (σ
unknown):
T =
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X̄ − µ0
√
S/ n
lOMoARcPSD|5093418
z test of hypothesis for a single
proportion:
z test for the difference between two means
(variances known):
P − π0
Z∼
=p
π0 (1 − π0 )/n
Z=
t test for the difference between two means
(variances unknown):
X̄1 − X̄2 − (µ1 − µ2 )
T = q
Sp2 (1/n1 + 1/n2 )
Confidence interval endpoints for the
difference between two means:
s

1
1
2
+
x̄1 − x̄2 ± tα/2, n1 +n2 −2 × sp
n1 n2
t test for the difference in means in
paired samples:
Pooled variance estimator:
Sp2 =
(n1 − 1)S12 + (n2 − 1)S22
n1 + n2 − 2
T =
Confidence interval endpoints for the
difference in means in paired samples:
P =
R1 + R 2
n1 + n2
χ2 statistic for test of association:
r X
c
X
(Oij − Eij )2
Eij
i=1 j=1
Spearman rank correlation:
6
rs = 1 −
n
P
Z=p
P1 − P2 − (π1 − π2 )
P (1 − P ) (1/n1 + 1/n2 )
Confidence interval endpoints for the
difference between two proportions:
s
p1 (1 − p1 ) p2 (1 − p2 )
+
p1 −p2 ±zα/2 ×
n1
n2
Sample correlation coefficient:
r = s
n
P
i=1
n
P
i=1
xi yi − nx̄ȳ
x2i − nx̄2
b=
1)
n
P
n
P
i=1
yi2 − nȳ 2
xi yi − nx̄ȳ
i=1
n
P
i=1
x2i − nx̄2
a = ȳ − bx̄
UL18/0322

Simple linear regression line estimates:
d2i
i=1
n(n2 −
X̄d − µd
√
Sd / n
z test for the difference between two
proportions:
sd
x̄d ± tα/2, n−1 × √
n
Pooled proportion estimator:
X̄1 − X̄2 − (µ1 − µ2 )
p
σ12 /n1 + σ22 /n2
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
UL18/0322
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
UL18/0322
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
UL18/0322
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
UL18/0322
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
UL18/0322
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
UL18/0322
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
UL18/0322
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
UL18/0322
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
UL18/0322
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
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lOMoARcPSD|5093418
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
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