Lia Liu
Stat 101 Review for Final Fall, 2015
1
Lia Liu
The final exam is on Wednesday, 1:00-3:00 pm. Room assignment will be posted later. You can bring two sheets of
formula and calculator. Final covers Chapter 15-19 only, is
worth 200 points. Use the following check list of concepts and
the Practice Final on your MyLab to prepare for the final.
Roughly speaking, the last unit is about the CLT and its
three applications:
1) From the sampling distribution, we learn for Binomial
model, the mean, variance and standard deviation of X (number of heads), sample proportion p̂ = Xn ; and for Normal model,
the mean, variance and standard deviation of the sample mean
X̄ and sample sum Sn .
2) CLT on the sample mean X̄ and sample sum Sn .
3) Application 1 of CLT: Estimate probability or use Empirical rule to describe a distribution, or decide whether an event
is unusual.
4) Application 2 of CLT: Find confidence interval for µ or p.
5) Application 3 of CLT: Do hypothesis testing for µ or p.
List of topics:
1. Sampling distribution: If X1 , X2 , , Xn is a random sample
from a population with mean µ and standard deviation σ,
n
then the sample mean X̄ = X1 +...+X
has mean E(X̄) = µ,
n
variance V (X̄) = σ 2 /n.
The sample sum Sn = X1 + ... + Xn has mean E(Sn ) = nµ, and
variance V (Sn ) = nσ 2 .
What about 2X1 −3X2 ? Can you find the mean and variance?
If X has binomial B(n, p), what are the mean and variance
of X? what are the mean and variance of p̂ = Xn ? How to
use normal to approximate probability? What conditions
must be checked?
Lia Liu
2
2. How to standardize a random variable X(find the z-score)?
Subtract by the mean of X, then divide by the standard
deviation of X.
X − E(X)
Z=
SD(X)
How to apply to ¯(X), p̂, X =the number of heads in Binomial
model?
3. Fundamental theorem of statistics (Central Limit Theorem): Apply to X̄, apply to p̂.
4. Use Empirical rule to describe a distribution, or estimate
probability or decide whether an event is unusual.
5. Confidence interval: estimate ± margin of error
6. How wide is a confidence interval?
7. How can you cut the width of a confidence interval by half
but keep the same confidence level?
8. Confidence interval for p (Binomial model): p̂ ± z
large enough.(When is n large enough?)
∗
r
p̂q̂
n
if n is
9. Confidence interval for µ (Normal model): (1) x̄ ± z ∗ √σn if
the standard deviation for the population, σ, is given;
(2) x̄ ± z ∗ √sn if the standard deviation for the population, σ,
is not given but n is large;
(3) x̄ ± t∗(n−1) √sn if the standard deviation for the population,
σ, is not given and n is small.
10. Hypothesis testing for p, µ: 1)know how to read a story and
write down the hypotheses. 2) which conditions to check; 3)
find P-value; 4) give conclusion, and interpret your conclusion
in context.
11. How to find p-values?
12. How to make a decision based on the p-value?
13. What is a Type I error?
Lia Liu
3
14. What is a Type II error?
15. What is the power of a test?
16. Can you reduce both Type I and Type II error at the same
time?
17. How is α and power related if n is fixed, if n is increasing,
if n is decreasing?