psych 218 - M2

Question 1

Types of probability

Answer 1

• ‘a priori’: before data / observation
  
  • based on expectations
  • p(A) = # of A-possible events / N-possible events
  • i.e. dice landing on an odd roll: p(A) = 3/6 = 0.5
• ‘a posteriori’: after data observation
  
  • based on what we observe
  • would not equal a priori, but should be relatively similar > will be skeptical if very different
  • p(A) = # of A-observed / N-observed
  • p(A) = 58 odd rolls / 100 rolls = 0.58
• have to decide if difference (0.50 and 0.58) is systematic or random factors
  
  • dice is fair / not enough evidence its unfair: maintain probability at 0.5
  • dice is unfair: predict probability at 0.58 for the future

Question 2

Probability terms

Answer 2

• mutually exclusive events (P & Q): events that cannot happen at the same time
  
  • i.e. dice cannot land on even and odd
  • P and Q will be complementary: P = 1 - Q
  • P and Q will never happen together: p(P & Q) = 0
• independent events: events that have no influence on each other
  
  • i.e. first dice roll does not influence second dice roll
  • correlation between independent events: p = 0.000
• exhaustive set of events: describe all possible events
  
  • i.e. even and not even dice roll
  • we know whether dice roll is “A” or “not A”

Question 3

Probability Rules

Answer 3

• addition rule: used when talking about one of several possible outcomes
  
  • p(A or B) = p(A) + p(B) - p(A and B)
  • if events are mutually exclusive, p(A and B) = 0
• multiplication rule: quantifies probability of successive events
  
  • independent successive events: i.e. probability of rolling sum of 11?
    
    • p(5) x p(6|5) = 1/6 x 1/6 = 1/36
    • p(6) x p(5|6) = 1/6 x 1/6 = 1/36
    • add up: 1/36 + 1/36 = 2/36
  • dependent successive events: i.e. probability of being dealt 2 Kings?
    
    • p(king) = 4/52
    • p(king | king) = 3/52
    • p(2 kings) = 4/52 x 3/52

Question 4

Goal of using inferential statistics?

Answer 4

• if we can assume population is normally distributed, we can predict where they fall
• can calculate the probability of sampling a certain z-score

Question 5

5 rules for binomial distribution

Answer 5

• series of N trials
  
  • multiple choice test with 20 questions
• only 2 outcomes
  
  • right v wrong answer
• outcomes are mutually exclusive
  
  • if you’re right, you cannot be wrong
• outcomes are independent
  
  • probability of solving Q1 is independent from Q2
• probability of P is consistent
  
  • always 25% because there are 4 options

Question 6

z-distribution (= normal distribution)

Answer 6

• shows how likely observations are when many outcomes are possible
• expectation = a priori probability
  
  • n x p (can be quantified before data collection)
  • there will be deviations from expectations (a priori ≠ a posteriori)
• as N gets bigger, distribution will look more and more normal

Question 7

Binomial v Normal Distribution

Answer 7

• as N increases, binomial distribution will look more and more like a normal distribution
  
  • can solve problems using z-scores and normal curves
  • as P and Q deviate from 0.50, binomial distribution looks less normal
• can only use normal distribution if:
  
  • N x P ≥ 10
  • N x Q ≥ 10

Question 8

Parameters of the normal distribution

Answer 8

• mean of distribution
  
  • μ = N x P
• standard deviation of distribution
  
  • σ = √ N x P x Q

Question 9

Example: Mosquito Preference

Answer 9

• N = 20, X = 15 bites
  
  • a priori: P(bite) = 0.5
  • a posteriori: P(bite) = 0.75 (15 out of 20 bites)
• assuming a priori probability, determine how strange the result is:
  
  • p(≥ 15) = 0.0148 + 0.0046 + 0.0011 + 0.0002 + 0.000 + 0.000 = 0.02
  • must add up all values more extreme to get area under the curve of a priori
• there is a 2% chance of X getting 15+ bites if null is true (= mosquito has no preference)
  
  • as 2% is very low, can conclude that the mosquito has preference

Question 10

How to Solve Binomial

Answer 10

• [1] Figure out what you are solving for
  
  • the probability you want to obtain (a priori)
  • evaluating the outcome we observe (a posteriori)
• [2] Consider outcomes that would be even more extreme
• [3] Look at the table
  
  • how many observations do we have (N)
  • probability of an event (a priori probability)
  • translate score into number of questions
    
    • i.e. if I need 80% to pass, need to solve at last 40 questions

Question 11

Normal Approximation

Answer 11

• converting the information we have into a z-score (can use normal curve)
  
  • 50 questions, need to get 80% score
  • p(correct) = 0.7 (based on student’s knowledge)
• z = (X - μ) / σ = 1.54
  
  • X: number of questions needed to get right
    
    • 50 x 0.8 = 40
  • μ = N x P
    
    • 50 x 0.7 = 35
  • σ = √ N x P x Q
    
    • √50 x 0.7 x 0.3 = 3.24

Question 12

Hypothesis Terms

Answer 12

• hypothesis: proposed explanation for an observation / phenomenon
• hypothesis testing: construct 2 hypothesis and have them compete with each other
  
  • must be mutually exclusive
  • must be exhaustive
• null hypothesis (Ho): any observed change is due to chance or unexplained factors
  
  • IV does not effect DV / no relationship / groups do not differ from each other
  • always start out assuming null hypothesis is true (because we can quantify it mathematically)
  • the extent we deviate from expectation: can decide if its due to randomness or something systematic
• alternative hypothesis (H1): observed change is not due to chance or unexplained factors
  
  • if non-directional, null will be “IV has no effect on DV”
  • if directional, null will be “IV does not have effect on DV, or effect is opposite to alternate”

Question 13

Hypothesis Testing

Answer 13

• reject null if
  
  • p < α
  • z-obt > z-crit
• fail to reject null if
  
  • p > α
  • z-obt < z-crit

Question 14

Statistical Significance

Answer 14

• we can claim statistical significance when we can reject null hypothesis
  
  • can rule out chance as the only reason why treatment group did better
  • chance is still a factor, but chance on its own is not able to account for the size of discrepancy – difference can be attributed to the treatment
• note: statistically significant does not indicate how meaningful treatment is
  
  • treatment group has lower fear levels, but we do not know strength or magnitude of the effect
  • thus, must calculate effect size to see extent
  • better to say “statistically reliable”

Question 15

Sign Test

Answer 15

• ignores magnitude, only considers sign (+ / -)
• requires repeated measures design
• steps:
  
  • [1] Check textbook for N number of events
  • [2] Check column for a priori probability
  • [3] Add up all extreme scores to see how likely it is to occur
    
    • if two-tailed, all extreme scores include scores on the other side of the distribution

Question 16

Power

Answer 16

• power (1-β): probability that the results of an experiment will allow rejection of the null if the IV has a real effect
  
  • sensitivity to detect a real effect
  • we want ≤ 80%
• useful to know power:
  
  • when designing an experiment
  • interpreting results when we retain null (to what extent is it statistically significant?)
    
    • with low power, we can never accept null because the experiment is not powerful enough to see an effect
    • have to consider power in interpretations – cannot assume null is correct
• relationships:
  
  • β: as β increases, power decreases (-)
  • N: as N increases, power increases (+)
  • α: as α increases, power decreases (-)

Question 17

Types of Errors

Answer 17

• Type I Error (α): decision to reject null hypothesis, when the null hypothesis is true
  
  • claim there is an effect, when there is indeed no effect
  • generally, worse in Psychology (spend time and money to investigate further in a relationship that has no effect)
• Type II error (β): decision to retain null hypothesis, when the null hypothesis is false
  
  • claim there is no effect, when there is indeed an effect
• minimizing α:
  
  • will decrease probability of Type I error, where p(correctly concluding) will be higher
  • but this will increase probability of Type II errors
• how to minimize β:
  
  • have large N when α is strict
  • use inference testing that is most powerful for the data
  • control conditions so variability of data is reduced

Question 18

Choosing Tails

Answer 18

• evaluation must always be two-tailed unless:
  
  • [1] there is no practical difference if results are the opposite direction (i.e. testing if new tires are faster – no point if new tires are slower)
  • [2] good theoretical reasons with strong supporting data
• have to maintain decision of one or two-tailed
  
  • switching from one to two-tailed inflates type I error

Question 19

z-test

Answer 19

• z-test: whether the sample mean is strange according to the null hypothesis
  
  • compares sample mean collected (x̄) and mean of known distribution (µ)
  • must know x̄ and µ
  • z = x̄ - µ / (σ / √N)
• variability of sample mean depends on how large the sample is
  
  • divide σ by √N is a correction
  • larger N will increase z because a larger sample reduces the influence of variability in a study
• steps:
  
  • [1] generate hypothesis: one or two-tailed
  • [2] set α level
  • [3] calculate standard error of the mean (σx̅)
  • [4] get z-obtained
  • [5] compare p v α / z-obt v z-crit

Question 20

Factors that change power

Answer 20

• increasing power
  
  • 1-tail test
  • larger α
  • increase N
  • manage σ through controlled studies
    
    • trade-off between statistical power and ecological validity
• reducing power
  
  • smaller mean differences (x̄ - µ)
  • larger variability (σx̅)

Question 21

3 ways to calculate power

Answer 21

• a priori: before data
  
  • need to know α
  • need to know power level we want (1-β)
  • need to know size of effect we expect to observe
    
    • we can never know how large effect of IV is before experiment
    • can estimate from pilot / past research
  • solves for N
• post hoc: after data (don’t do it due to circular reasoning)
  
  • need to know α
  • need to know N
  • need to know size of effect we expect to observe
  • solves for power (1-β)
• sensitivity: before / after data
  
  • need to know α
  • need to know power level we want (1-β)
  • need to know N
  • solves for size of effect

Question 22

Sampling Distribution of the Mean

Answer 22

• sampling distribution of the mean: all the values the mean can take, along with the probability of getting each value if sampling is random from a null hypothesis population
  
  • on average, sample means will be centred at population mean
• each sample mean is an estimate of µ
  
  • sample mean = best guess of population mean
• properties of sample means:
  
  • mean of distribution of sample means = population mean
  • standard deviation of distribution of sample means = population standard deviation / √N

Question 23

Standard Error of the Mean

Answer 23

• standard error of the mean: mean difference between x̄ and µ
• depends on N: more people = less likely sample is biased

Question 24

t-test

Answer 24

• used when we don’t have all the components for z-test (µ and σ)
• steps
  
  • [1] set a hypothetical µ
    
    • assume null is true
  • [2] estimate σ
    
    • assume s = σ

Question 25

degrees of freedom

Answer 25

- assuming s = σ is biased because we will underestimate σ - will think there is less variability than there actually is - correct using degrees of freedom - done through formula for s that includes "N - 1" - as degree of freedom increases, peak gets higher

Question 26

z-test v t-test

Answer 26

- z-distribution is always normal, t-distribution approaches normal as N increases - t-test estimates σ from s