Final Exam Review Chapter 10 Know the three ideas of sampling. • Examine a part of the whole: A sample can give information about the population. vA parameter is a number used in a model of the population. vA statistic is a number that is calculated from the sample data. vThe sample to sample differences are called the sampling variability (or sampling error). • Randomize to make the sample representative. • The sample size is what matters. • In a simple random sample (SRS), every possible group of n individuals has an equal chance of being our sample. Chapter 13 • Know the general rules of probability and how to apply them. • The General Addition Rule says that : P(A) or P(B) = P(A) + P(B) – P(A and B). • The General Multiplication Rule says that : P(A and B) = P(A) x P(B|A). •Know that the conditional probability of an event B given the event A is P(B|A) = P(A and B)/P(A). •Know how to define and use independence: Events A and B are independent if P(A|B) = P(A) or P(A and B) = P(A) × P(B) Chapter 14 • The expected value of a (discrete) random variable is: • The variance for a random variable is: • Rules combining RVs: E(X ± c) = E(X) ± c Var(X ± c) = Var(X) E(aX) = aE(X) Var(aX) = a2Var(X) ( ) ( )E X x P xµ = = ⋅∑ ( ) ( ) ( )22 Var X x P xσ µ= = − ⋅∑ Chapter 15 :Geometric Probability Model for Bernoulli Trials: GEOM(n, p) •p = probability of success •q = 1 – p = probability of failure •X = number of trials until the first success occur •Expected value: •Standard deviation: P X = x( ) = qx−1p E X( ) = µ = 1 p σ = q p2 Chapter 15: Binomial Probability Model for Bernoulli Trials: BINOM(n, p) •x = number of trials •p = probability of success •q = 1 – p = probability of failure •X = number of successes in n trials P X = x( ) = nCx pxqn−x, nCx = n! x! n − x( )! Mean: µ = np Standard deviation: σ = npq Chapter 15:Poisson for Small p • For rare events (small p), np may be less than 10. • Use the Poisson instead of the Normal model. • l = np mean number of successes • X = number of successes • • • Good approximation if n ³ 20 with p ≤ 0.05 or n ≤ 100 with p ≤ 0.10 ( ) ! ll- == xe P X x x ( ) , ( )l l= =E X SD X Chapter 16: One-Proportion Z-Interval • Conditions met, find level C confidence interval for p • Confidence interval is • Standard deviation estimated by • z* specifies number of SEs needed for C% of random samples to yield confidence intervals that capture the true parameter. Use table below to get z* • Interpretation : we are 95% confident that the interval contains the true proportion of X in the population p̂ ± z * ×SE(p̂) SE(p̂) = p̂q̂ n Chapter 16: One-Proportion Z-Interval •Sampling Distribution for Proportions is Normal. • Mean is p. • σ (p̂) = SD(p̂) = pq n Chapter 16: One-Proportion Z-Interval Sample Size and Standard Deviation • • Larger sample size → Smaller standard deviaaon σ ( )=SD y n ˆ( )= pq SD p n Chapter 16: One-Proporti ...

1. Final Exam Review
Chapter 10
Know the three ideas of sampling.
• Examine a part of the whole: A sample can give
information about the population.
vA parameter is a number used in a model of the population.
vA statistic is a number that is calculated from the sample data.
vThe sample to sample differences are called the sampling
variability (or sampling error).
• Randomize to make the sample representative.
• The sample size is what matters.
• In a simple random sample (SRS), every possible group of
n individuals has an equal chance of being our sample.
Chapter 13
• Know the general rules of probability and how to apply them.
• The General Addition Rule says that :
P(A) or P(B) = P(A) + P(B) – P(A and B).
• The General Multiplication Rule says that :
P(A and B) = P(A) x P(B|A).
•Know that the conditional probability of an event B given the
event A is P(B|A) = P(A and B)/P(A).
•Know how to define and use independence:
Events A and B are independent if P(A|B) = P(A) or P(A and B)

2. =
P(A) × P(B)
Chapter 14
• The expected value of a (discrete) random variable is:
• The variance for a random variable is:
• Rules combining RVs:
E(X ± c) = E(X) ± c Var(X ± c) = Var(X)
E(aX) = aE(X) Var(aX) = a2Var(X)
( ) ( )E X x P xµ = = ⋅ ∑
( ) ( ) ( )22 Var X x P xσ µ= = − ⋅ ∑
Chapter 15 :Geometric Probability Model
for Bernoulli Trials: GEOM(n, p)
•p = probability of success
•q = 1 – p = probability of failure
•X = number of trials until the first success occur
•Expected value:
•Standard deviation:
P X = x( ) = qx−1p
E X( ) = µ = 1
p
σ =

3. q
p2
Chapter 15: Binomial Probability Model for
Bernoulli Trials: BINOM(n, p)
•x = number of trials
•p = probability of success
•q = 1 – p = probability of failure
•X = number of successes in n trials
P X = x( ) = nCx pxqn−x, nCx =
n!
x! n − x( )!
Mean: µ = np
Standard deviation: σ = npq
Chapter 15:Poisson for Small p
• For rare events (small p), np may be less than 10.
• Use the Poisson instead of the Normal model.
• l = np mean number of successes
• X = number of successes
•
•
• Good approximation if n ³ 20 with p ≤ 0.05

4. or n ≤ 100 with p ≤ 0.10
( )
!
ll-
==
xe
P X x
x
( ) , ( )l l= =E X SD X
Chapter 16: One-Proportion Z-Interval
• Conditions met, find level C confidence interval for p
• Confidence interval is
• Standard deviation estimated by
• z* specifies number of SEs needed for C% of random
samples to yield confidence intervals that capture the
true parameter. Use table below to get z*
• Interpretation : we are 95% confident that the interval contains
the
true proportion of X in the population
p̂ ± z * ×SE(p̂ )
SE(p̂ ) =

5. p̂ q̂
n
Chapter 16: One-Proportion Z-Interval
•Sampling Distribution for Proportions is Normal.
• Mean is p.
•
σ (p̂ ) = SD(p̂ ) =
pq
n
Chapter 16: One-Proportion Z-Interval
Sample Size and Standard Deviation
•
• Larger sample size → Smaller standard deviaaon
σ
( )=SD y
n
ˆ( )=
pq

6. SD p
n
Chapter 16: One-Proportion Z-Interval
Chapter 17: The Central Limit Theorem
•The Central Limit Theorem
• The sampling distribution of any mean becomes nearly Normal
as
the sample size grows.
•Requirements
• Independent
• Randomly collected sample
•The sampling distribution of the means is close to Normal if
either:
• Large sample size
• Population close to Normal
Chapter 17 : A Practical Sampling Distribution
Model
•When certain assumptions and conditions are met, the
standardized sample mean,
follows a Student’s t-model with n – 1 degrees of
freedom. We estimate the standard deviation with

7. t =
y − µ
SE y( ),
SE y( ) = s
n
.
Degrees of Freedom
• For every sample size n there is a different Student’s t
distribuaon.
• Degrees of freedom: df = n – 1.
• Similar to the “n – 1” in the formula for sample standard
deviaaon
• It is the number of independent quanaaes leh aher we’ve
esamated the
parameters.
One Sample t-Interval for the Mean
When the assumptions are met (seen later), the confidence
interval for the mean is
The critical value depends on the confidence level, C, and

8. the degrees of
freedom n – 1.
y ± tn−1
* × SE(y )
tn−1
*
Use R to get : abs(qt(0.05/2,df)) tn−1
*
Chapter 18 :1-Proportion z-Test
•Conditions
• Same as a 1-Proportion z-Interval
•Null Hypothesis
• H0: p = p0
•Test Statistics
• ( )
0
ˆ
ˆ
p p
z
SD p
-

9. = ( ) 0 0ˆ p qSD p
n
=
• Use pnorm(z*) to calculate area to the left of z*; 1-pnorm()
calculates the area to the right of z*
If your alternative hypothesis is p different than p0, you can use
2*(1-pnorm(z*)). This probability is the
p-value, and we reject the Null Hypothesis if p-value<0.05
Chapter 18: One-Sample t-Test for the Mean
• Assumpaons are the same.
• H0:
•
• Standard Error of
• When the condiaons are met and H0 is true, the staasac
follows the Student’s
t Model with n – 1 df.
• Use pt(t*,df) to calculate area to the leh of t*
µ = µ0
tn−1 =
y − µ0
SE(y )

10. y : SE(y ) =
σ
n
Chapter 18
• Write clear summaries to interpret a confidence interval or
state a hypothesis test’s conclusion.
• Understand P-values.
• A P-value is the estimated probability of observing a statistic
value at
least as far from the (null) hypothesized value as the one we
have
actually observed.
• A small P-value indicates that the statistic we have observed
would be
unlikely were the null hypothesis true. That leads us to doubt
the null.
• A large P-value just tells us that we have insufficient evidence
to doubt
the null hypothesis. In particular, it does not prove the null to
be true.
Chapter 19: Type I and II Errors
•Type I Error
• Reject H0 when H0 is true.

11. •Type II Error
• Fail to reject H0 when H0 is false.
•Medicine: Such as an AIDS test
• Type I Error → False positive: Patient thinks he has the
disease when he
doesn’t.
• Type II Error → False negative: Patient is told he is disease -
free when in fact
he has the disease.
Chapter 19 Probabilities of Type I and II Errors
• P (Type I Error) = a
• This represents the probability that if H0 is true then we will
reject H0.
• P (Type II Error) = b
• We cannot calculate b. Saying H0 is false does not tell us what
the parameter
is.
• Decreasing a results in an increase of b: By decreasing a, we
fail
to reject more ( it is harder to reject, e.g. p-value=0.04 we
reject
at a=0.05)
• Reduce b for all alternatives, by increasing a.

12. • The only way to decrease both is to increase the sample size.
Chapters 20, 21 and 25: A Two-Proportion z-
Test
• H0: p1 – p2 = 0
•
•
• When Condiaons are met and the null hypothesis is true,
• This staasac follows the Normal model, which we can use to
obtain a P-
value.
• Use pnorm() to calculate the p-value
SE p̂ 1 − p̂ 2( ) =
p̂ 1q̂ 1
n1
+
p̂ 2q̂ 2
n2
z =
(p̂ 1 − p̂ 2 ) − 0
SE p̂ 1 − p̂ 2( )

13. Chapters 20, 21 and 25: A Two-Sample t-Test
for the Difference Between Means
• H0: µ1 – µ2 = D0 (D0 usually 0)
•
• When the conditions are met and the null hypothesis is true,
use the
Student’s t-model to find the P-value.
• Degrees of freedom will be given !
• Use pt() to calculate the p-value
t =
y1 − y 2( )− Δ0
SE(y1 − y 2 )
SE(y1 − y 2 ) =
s1
2
n1
+
s2
2
n2

14. Chapters 20, 21 and 25: One Way ANOVA F-
Test
• H0: µ1 = µ2 = … µk
•
• MST is found from the variance of the means of the treatment
groups.
• MSE is found by pooling the variances within each of the
treatment groups.
• If F is large, reject H0.
• Use aov() to get the p-value
T
E
MS
F
MS
=
Testing Hypothesis
• Reject Null Hypothesis if p-value<0.05
• For proportions use Normal Model (pnorm())
• For means use t-student Model (pt())
• To test if many means are equal use the F-statistic (aov())