1. Sampling distributions and
Hypothesis Testing
population/ sample/ sampling
parameter = µ, σ, ρ (greek letter, population)
statistic = x̅, s, r, m (latin letter, sample,
sampling distribution)
- the larger the sample, the closer the
sample mean x̅ to population mean µ
hypothesis testing
hypothesis = statement about parameter in
population
null hypothesis = H0 states nothing going on
(no change, no difference, no relation)
alternative hypothesis = Ha something going
on
hypothesis testing
1. hypothesis (statement about population
parameters h0 and ha)
- one- (<, >) and two-sided hypotheses
(=, ≠)
2. sampling distribution
3. test statistic (calculate from sample data to
compare sample statistics)
4. rejection region & p-value
- critical value = boundry value
- rejection region = area
- significance level α = percentage
- p-value = probability
5. statistical conclusion = significance? (H0
reject/true)
- two methods (compare p to α, compare
test statistic to critical value)
- p-value = p ≤ α H0 rejected (lower p-
value - stronger evidence)
- Fisher: p-value shows strength of
evidence against null-hypothesis
- Neyman-Pearson: can decide whether
null-hypothesis or alternative is better
6. substansive conclusion = answer
research question
errors
type I (probability α) = reject H0 when it is
true (conclude effect, but non)
type II (probability β) = failing to reject H0
when it is false (miss effect)
, 2. Basic Concepts of Probability
Random variables Probability
- variables with numerical values - relative frequency after infinite
obtained by sampling or any other observations
random process (values outcome of
random phenomena) Independent events
- discrete and continuous - occurance of A does not alter
probability of occurence of B
Law of large numbers - methods to determine independence
- the larger the sample, the closer the of 2 variables (yes- independent)
sample mean x̅ to population mean µ 1. are conditional probabilities equal
for all conditions?
Expected value 2. are conditional probabilities equal
µx = ∑ pixi to marginal probabilities?
- idealized population mean (all
answers) Disjoint events
- mutually exclusive (cannot occur at the
Compare: sample mean same time)
x̅ = ∑xi/n
Law´s of probability
Variance Additive law OR (probability that either
o2y = ∑ pi(xi - µx)2 happens)
- general: P(A or B)= P(A) + P(B) - P(A
Compare: sample variance and B)
∑ (xi - x̅)2/ n -1 - disjoint: P(A or B)= P(A) + P(B)
Algebra Multiplicative law AND (probability that both
Transformation of 1 variable X -> Y happen)
Add constant a (y = a + x) - general: P(A and B) = P(A) x P(B|A)
- new expected value: µy = a + x probability of A happening given B has
- new variance: o2y = o2x occurred
- independent: P(A and B) = P(A) x
Multiply by b (y = bx) P(B) probability that both events occur,
- new expected value: µy = bµx one does not change the probability of
- new variance: o2y = b2o2x the other
Linear transformation (y = a + bx) Complement law P(A) = 1 - P(AC)
- new expected value: µy = a + bµx
- new variance: o2y = b2o2x Probability Types
Marginal probability YES or NO
Sum of 2 random variables X + Y - probability for one variable
ex. Celcius -> Fahrenheit (F = 32 + 1.8C) - simple or unconditional probability
- new expected value: µx+y = µx + µy
- new variance: o2x+y = o2x + o2y + 2px ox Conditional probability GIVEN that
oy - probability for A given that B has
(depends on correlation rho between X occurred
and Y) - P(B|A) = P(A and B)/ P(A)
Joint probability AND
- probability of co-occurence of multiple
events
- general multiplicative law P(A and B) =
P(A) x P(B|A)
Hypothesis Testing
population/ sample/ sampling
parameter = µ, σ, ρ (greek letter, population)
statistic = x̅, s, r, m (latin letter, sample,
sampling distribution)
- the larger the sample, the closer the
sample mean x̅ to population mean µ
hypothesis testing
hypothesis = statement about parameter in
population
null hypothesis = H0 states nothing going on
(no change, no difference, no relation)
alternative hypothesis = Ha something going
on
hypothesis testing
1. hypothesis (statement about population
parameters h0 and ha)
- one- (<, >) and two-sided hypotheses
(=, ≠)
2. sampling distribution
3. test statistic (calculate from sample data to
compare sample statistics)
4. rejection region & p-value
- critical value = boundry value
- rejection region = area
- significance level α = percentage
- p-value = probability
5. statistical conclusion = significance? (H0
reject/true)
- two methods (compare p to α, compare
test statistic to critical value)
- p-value = p ≤ α H0 rejected (lower p-
value - stronger evidence)
- Fisher: p-value shows strength of
evidence against null-hypothesis
- Neyman-Pearson: can decide whether
null-hypothesis or alternative is better
6. substansive conclusion = answer
research question
errors
type I (probability α) = reject H0 when it is
true (conclude effect, but non)
type II (probability β) = failing to reject H0
when it is false (miss effect)
, 2. Basic Concepts of Probability
Random variables Probability
- variables with numerical values - relative frequency after infinite
obtained by sampling or any other observations
random process (values outcome of
random phenomena) Independent events
- discrete and continuous - occurance of A does not alter
probability of occurence of B
Law of large numbers - methods to determine independence
- the larger the sample, the closer the of 2 variables (yes- independent)
sample mean x̅ to population mean µ 1. are conditional probabilities equal
for all conditions?
Expected value 2. are conditional probabilities equal
µx = ∑ pixi to marginal probabilities?
- idealized population mean (all
answers) Disjoint events
- mutually exclusive (cannot occur at the
Compare: sample mean same time)
x̅ = ∑xi/n
Law´s of probability
Variance Additive law OR (probability that either
o2y = ∑ pi(xi - µx)2 happens)
- general: P(A or B)= P(A) + P(B) - P(A
Compare: sample variance and B)
∑ (xi - x̅)2/ n -1 - disjoint: P(A or B)= P(A) + P(B)
Algebra Multiplicative law AND (probability that both
Transformation of 1 variable X -> Y happen)
Add constant a (y = a + x) - general: P(A and B) = P(A) x P(B|A)
- new expected value: µy = a + x probability of A happening given B has
- new variance: o2y = o2x occurred
- independent: P(A and B) = P(A) x
Multiply by b (y = bx) P(B) probability that both events occur,
- new expected value: µy = bµx one does not change the probability of
- new variance: o2y = b2o2x the other
Linear transformation (y = a + bx) Complement law P(A) = 1 - P(AC)
- new expected value: µy = a + bµx
- new variance: o2y = b2o2x Probability Types
Marginal probability YES or NO
Sum of 2 random variables X + Y - probability for one variable
ex. Celcius -> Fahrenheit (F = 32 + 1.8C) - simple or unconditional probability
- new expected value: µx+y = µx + µy
- new variance: o2x+y = o2x + o2y + 2px ox Conditional probability GIVEN that
oy - probability for A given that B has
(depends on correlation rho between X occurred
and Y) - P(B|A) = P(A and B)/ P(A)
Joint probability AND
- probability of co-occurence of multiple
events
- general multiplicative law P(A and B) =
P(A) x P(B|A)
