Math 1180
Summary of topics and practice problems for Midterm 3
1. Maximum likelihhod estimators (in binomial, poisson, normal, geometric and exponential distributions), 8.1
2. Find confidence limits from the definition, 8.2.1
3. Estimates and Normal-based confidence limits for the mean, 8.3.1, 8.3.2
4. Computing sample variance, 8.3.3
5. Confidence limits for the mean using sample variance and either Normal
or t-distribution, 8.3.3, 8.3.4
6. Find confidence limits for binomial proportion from Normal approximation, 8.3.5
7. Hypothesis testing (formulate hypothesis and alternative, compute Pvalue, make conclusions (significance, reject/accept, word conclusion)
- from definition (8.4)
- With Normal approximatoin for means or proportions (8.5.1)
- With t-distribution (8.5.2)
- Comparing means or proportions with Normal approximation or tdistribution (8.6.1-8.6.3)
- Testing abainst a known baseline with chi-squared (8.7.1)
- Test for independence with chi-squared (8.7.2)
8. Linear regression (8.9)
There are no extra problems this time. We will focus only on
homework problems.
Formulae list
If you think some formulae are missing, please let me know!
X̄ = E(X) = Σni=1 xi p(xi )
Z b
X̄ = E(X) =
xf (x)dx
a
n k
p (1 − p)n−k
P r(N = k) = b(k; n, p) =
k
E(N) = np, V ar(N) = np(1 − p)
P r(T = t) = gt = p(1 − p)t−1
P r(T ≤ t) = Gt = 1 − (1 − p)t
1
1−p
E(T ) = , V ar(t) =
p
p2
(λt)k e−λt
k!
E(N) = λt, V ar(N) = λt
P r(N = k) = p(k; λ, t) =
T : f (t) = λe−λt ; F (t) = 1 − e−λt
1
1
E(T ) = , V ar(T ) = 2
λ
λ
Sn ∼ N(nµ, nσ 2 )
An ∼ N(µ, σ 2 /n)
X −µ
σ
(x−µ)2
1
e− 2σ2
f (x) = √
2πσ 2
Z=
L(p) = P r(Data if parameter is equal to p)
P r(result greater than or equal to data if p = pl ) =
P r(result less than or equal to data if p = ph ) =
α
2
α
2
σ
ml,h = µ̂ ± 1.96 √
n
Pn 2
x − nµ̂2
s2 = i=1 i
n−1
s
pl,h = p̂ ± 1.96 √
n
s2 = p̂(1 − p̂)
P − value = Probability to observe a result at least as extreme as the data
if the null hypothesis is true
χ2 =
m
X
(Oi − Ei )2
i=1
Ei
Normal table from the end of the book
t-distribution table from page 739
table for chi-squared page 775