Objectives (BPS chapter 12)

General rules of probability

1. Independence : Two events A and B are independent if the probability that

one event occurs is not influenced by the occurrence of the other event.

2. multiplication rule for independent events: P(A and B) = P(A) P(B)

3. The general addition rule: P(A or B) = P(A) + P(B) – P(A and B)

4. Conditional probability:

5. General multiplication rule: P(A and B) = P(A)P(B|A)

6. If A and B are independent, then P(A and B) = P(A)P(B)

)()()|(

APBandAPABP

Objectives (BPS chapter 13)Binomial distributions

1. The binomial setting: Fixed number of trials, probability of success

2. Binomial distributions B(n, p): models for some categorical variables, typically

representing the number of successes in a series of n trials.

3. Binomial probabilities:

4. Binomial mean and standard deviation:

5. The Normal approximation to binomial distributions: If X is the count of

successes in the sample and np ≥10 and n(1 − p) ≥10, the sampling

distribution for large sample size n is:

knk ppnkkXP

)1()(

)1( pnp

np

))1(,(~ pnpnpNX

Objectives (BPS chapter 14)

Confidence intervals: σ known

1. C% Confidence intervals for the mean when population standarddeviation,σ known.

2. Choosing the sample size

x z * n

m z * n

n z *

m

2

Objectives (BPS chapter 15)1. Tests of significance: σ known

Step 1: Stating hypothesesH0 : the statement being tested, a statement of “no effect” or “no difference”. Ha: the claim we are trying to find evidence for

Step 2: Calculate Test statistics

Step 3: Obtain P-values using Table A

Step 4: If the P-value is equal to or less than α (p ≤ α), then reject H0. If the P-value is greater than α (p > α), then fail to reject H0.

2. Tests from confidence intervals: Because a two-sided test is symmetrical, you can also use a confidence interval to test a two-sided hypothesis.

nxz

0

Objectives (BPS chapter 18)Inference about a Population Mean: σ unknown

1 The one-sample t confidence interval

2. The one-sample t test :

3. Use Matched pairs t test if two samples are not independent

nstx *

nsxt

nsxt 0

H0: difference = 0 ;Ha: difference > 0 (or <0, or ≠0)

Objectives (BPS chapter 19)Comparing two population means: σ1, σ2 unknown

1. Degree of freedom, df= smallest (n1−1; n2−1)

2.Two sample t-confidence interval

3. Two-sample t-test

2

22

1

21

21 *)(ns

nstxx

t x 1 x 2s1

2

n1

s22

n2

Objectives (BPS chapter 20)Inference for a population proportion

1. The sample proportion

2. The sampling distribution of

3. Large sample confidence interval for p

4. Choosing the sample size

5. Significance tests for a proportion

p

npppNp )1(,~

nppzp )ˆ1(ˆ*ˆ

*)1(** 2

ppmzn

npp

ppz)1(

ˆ

00

0

Objectives (BPS chapter 21)Comparing two proportions

1. The sampling distribution of a difference between proportions

2. Large Sample confidence intervals for comparing two proportions

3. Significance tests for comparing proportions

2

22

1

112121

)1()1(,~)ˆˆ(n

ppn

ppppNpp

2

22

1

1121

)ˆ1(ˆ)ˆ1(ˆ*)ˆˆ(

npp

nppzpp

21

210

::

ppHpppH

a

21

21

11)1(ˆnn

pp

ppz

21

21

2

22

1

11

ˆ

ˆ

ˆ

nnxxp

nxp

nxp

Objectives (BPS chapter 23)The chi-square test

1. Chi-square hypothesis test

H0: There is no relationship between categorical variable A and B.

Ha: There is some relationship between categorical variable A and B.

2. Expected counts in two-way tables

3. The chi-square test

Obtain p-value using Table E with degree of freedom, df=(r-1)(c-1)

4. Cell counts required for the chi-square test

All individual expected counts are 1 or more (≥1)

No more than 20% of expected counts are less than 5 (< 5)

count expected

count expected -count observed 2

2

Objectives (BPS chapter 24)Inference for regression

1. Testing the hypothesis of no linear relationship

H0: = 0 vs. Ha: ≠ 0, >0, or <0

Find the p-value using Table C with degree of freedom, df=n-2

2. Confidence intervals for the regression slope:

3. prediction interval for a single observation when x=x*:

4 confidence interval for the mean response when x=x*:

)( 2

xxs

bSEbt

b

bSEtb *

ySEty ˆ*

*SEty