T-tests

Question 1

The Principle of Hypothesis Testing

Answer 1

Start with a default assumption, the null, and ask whether the evidence from the sample is strong enough to reject it

Question 2

Null Hypothesis

Answer 2

H0: μ = 50, for example

Question 3

Alternative Hypothesis

Answer 3

H1: μ ≠ 50, for example

reject or retain H0

Question 4

α (level of significance)

Answer 4

the amount of error the experimenter is willing to tolerate.

when estimating a parameter from a sample statistic it is not possible to eliminate error, only to quantify the likelihood of error

typically set at .05 if not otherwise stated

Question 5

Decision Rule

Answer 5

Reject H0 if the probability of obtaining a sample mean as deviant or more deviant than the one observed, when H0 is true, is less than or equal to α

convert M to zo first, z = M - μ0 / σM

Decision rule: Reject H0 if |Z| ≥ Zc

Question 6

Decision Rule and Deciding α

Answer 6

Considering the decision rule changes what we consider to reject or retain by the mean, α becomes a criterion for rejecting or retaining H0

you might think why not choose .01 to minimise the most errors, well, there are multiple types of errors

i.e. we want to strike a balance between type 1 and 2 errors

Question 7

Type 1 Error

Answer 7

Rejecting H0 when H0 is true

Question 8

Correct Acceptance

Answer 8

Retain H0 when H0 is true

Question 9

Correct Rejection

Answer 9

Reject H0 when H0 is false

Question 10

Type II Error

Answer 10

Retain H0 when H0 is false

Question 11

Which Error is regarded as worse?

Answer 11

Type 1 errors, so the emphasis is on controlling for this. People almost always set α at .05 or less

Question 12

Degrees of Freedom

Answer 12

the number of independent observations in a set of data

Question 13

Principle of T-tests

Answer 13

When we don’t know the population mean (μ)

Question 14

Note on sM (instead of σM)

Answer 14

there is no longer a normal distribution using the sample

Question 15

Properties of t distribution

Answer 15

as df increases, t distribution becomes closer in shape to a normal distribution

• for lower values of df, t distribution has narrowed peak and fatter tails than normal distribution, and therefore has a larger standard deviation

Question 16

Confidence Intervals for μ (σ unknown)

Answer 16

μ upper = M + (tc x sM)

μ lower = M - (tc x sM)

Question 17

T-test formula for a single mean

Answer 17

√ Σ(X-M)^2 / n (n-1)

Question 18

Paired Samples Design for Two Means

Answer 18

same participant on multiple occasions, measuring one participants change over time (within groups)

test of dependent means

Question 19

Independent Samples Design for Two Means

Answer 19

cases are independent from each other (between-groups)

independent means

separate people placed in different groups

Question 20

Difference Score Method

Answer 20

X1 - X2 = XD

XD - MD

Question 21

Paired Samples Equation

Answer 21

sMD = √Σ(XD-MD)^2 / n(n-1)

Question 22

Confidence Interval for Paired samples

Answer 22

uD upper = MD + (tc x sMD)

uD lower = MD + (tc x sMD)

we can be 95% that population mean … is between … and … under … condition than … condition

Question 23

P-value

Answer 23

the probability of obtaining a t statistic that is deviant to the one obtained, even when the null is true (under h0)

probability that the result occurred by chance when there was no true effect

p<0.05 could be considered statistically significant

Question 24

Natural Pairs

Answer 24

-The researcher does not randomly allocate participants to one group or another; the pairing occurs ‘naturally’ prior to the study

• Although the pairs are made up of different participants the scores of each pair are likely to be correlated → dependent means variable
• Example: researcher wants to compare marital satisfaction of husbands and wives

Question 25

Matched Pairs

Answer 25

- The researcher has control over the ways the pairs are formed so that participants are matched on some variable - Although the pairs are made of different participants the scores of each pair are correlated → dependent means analysis - Example: examine effects of stress on food intake in rats, participants divided into pairs matched on weight, and within pairs are randomly allocated to groups (stressed and non-stressed group), two groups won't have a third variable for weight

Question 26

Standard Error Formula when finding difference between means (M1-M2)

Answer 26

σ M1-M2 = √ σ1^2/n1 + σ 2^2/n2

Question 27

Formula for Independent Samples Mean

Answer 27

S M1-M2 =√Σ(X1-M1)^2 +Σ(X2-M2)^2 / (n1-1) +(n2-1) x (1/n1 + 1/n2)

Question 28

Independent Scores Method

Answer 28

X1-M1 + X2-M2

Question 29

Confidence Intervals for (μ1-μ2)

Answer 29

(μ1-μ2) upper = (M1-M2) + (tc X S m1-m2) (μ1-μ2) lower = (M1-M2) - (tc X S m1-m2)

Question 30

Finding t for Independent Means

Answer 30

t = (M1-M2) / Sm1-m2

Question 31

Assumptions for Independent Means T-tests

Answer 31

1. Normal distribution of scores in each population 2. Population variances are equal 3. Observations are independent, within and between groups (one person's productivity while listening to music does not influence another person's productivity while listening to music)

Question 32

Changes in Outcomes if Independent t-test assumptions are met or not

Answer 32

- When all 3 met, Type 1 error rate = nominal rate (a) (the rate expected by the experimenter) - When assumption are violated, actual distribution of t values does not follow theoretical t distribution - As a result, actual type 1 error rate does not equal a

Question 33

Robustness

Answer 33

a test is said to be robust against violation of an assumption if the actual type 1 error rate is close to α even when that assumption is violated - evaluated using monte carlo simulations

Question 34

What is a t-test robust against?

Answer 34

violation of normality assumption - central limit theorem - sampling distribution will always be normal if the sample size is large enough violation of assumptions of equal variances provided n1=n2 - when unequal outcomes depends on which group has larger variance - When the larger sample is associated with larger variance, the t-test is conservative (think you have a big standard error when you actually don’t) (less type 1 errors, but higher risk of type 2 errors) - When the larger sample is associated with the smaller variance, the t-test is liberal (more type 1 errors, less risk of type errors) (type 1 error rate will go above alpha) (type 1 errors are more important)

Question 35

What is the t-test not robust against?

Answer 35

Violation of independence of observations (when the data points are unrelated to each other - groups tested together, such as comparing 2 classrooms - Will not be a good experiment, naturalistic, quasi-experiment - Here the behaviour of one participant may influence the behaviour of other participants - May be a very good or bad student that may have an influence on all the scores in that condition - Classes may not be equal in their academic ability etc

Question 36

EV's

Answer 36

extraneous variables - any variable that can effect the dependent variables of your study intrinsic - participant variables - temporary EV's (mood, motivation, etc) - relatively stable EV's (intelligence, ability, personality) other EV's relate to the environment - testing conditions, measurement error

Question 37

Are EV's relative?

Answer 37

one researchers EVs may be another researchers IVs

Question 38

Controlling for EV's

Answer 38

- Internal validity - an experiment is internally valid if there are no other explanations for changes in the DV, other than the manipulation of the IV EV’s are influences on the DV not due to the IV, hence threaten internal validity - Need to - Make sure the only systematic difference between conditions is the IV - Reduce amount of error in data due to EVs i.e through group variability

Question 39

Random Allocation for Independent Samples

Answer 39

Controlling for EV's related to participants - Random allocation is used to convert possible systematic errors into random errors - Provides some control over all potential participant EVs, random allocation deals with them whether we know about it or not - Does not equate groups; rather random allocation distributions EVs impartially between groups Rather than random sampling, which increase external validity, as we get a diverse and impartial population base

Question 40

Controlling EV's for Paired Samples

Answer 40

- Measuring same participants (repeated measure) or matched pairs produces a paired sample design - Ensures that distribution of scores on EVs related to participants are held constant from one condition to the next (individual differences are not applicable, because they are the same for each condition, because it is the same person) - This can reduce the standard error for the difference between conditions - Manages the variability of participant performance by measuring the difference between conditions for each individual participant

Question 41

Issue of Rank-Order Effects for EV's in Paired Samples

Answer 41

Rank order effects: conditions presented earlier may be responded to differently than conditions presented later - Positive rank order effects: practice, learning, reduction in symptom - Negative rank order effects: fatigue, boredom - These are problematic if conditions are confounded with order (everyone does condition X first and Y second - Solved with Counterbalancing - E.g if there are two conditions A and B, Half do A first B second, Half do B first A second - Random permutations: each participant does a random order of conditions

Question 42

Issue of Carry Over Effects in EV's for Paired Samples

Answer 42

- Effect of one condition carry to the next - Does not necessarily have to do with a specific order, only one condition - Not solved by counterbalancing

Question 43

Comparing Strength of Paired VS Independent

Answer 43

In paired samples, difference scores are normally lower in variance, correlation between scores in condition 1 and condition 2 should be positive, less random variability than independent sample - Standard error is smaller than when samples are independent - Therefore, we get a larger t - Thus, we are more likely to be rejected with paired samples design than independent - Dependent t test is a more powerful test of accepting the null than independent i.e more likely to lead to rejection of a false H0