Analysing data

Question 1

Greek symbols

Answer 1

population mean: µ
sample mean: ̅x
population mean estimate: μ ̂
SD: o-

Question 2

normal distribution

Answer 2

• bell curved
• peak is its mean
• mean median mode same value
• centring; changing mean, shifting curve left/right
• SD determines steepness of curve
• scaling; changing SD
• 68.2% of data within +/- 1 SD of mean
• 95.4% of data within +/- 2 SD of mean
• 99.7% of data within +/- 3 SD of mean

Question 3

critical values

Answer 3

if sd is known, can calculate critical value fro any proportion of normally distributed data

Question 4

sampling from distributions

Answer 4

• collecting data on variable includes randomly sampling from distribution
• underlying distribution assumed to be normal
• some variables may come from other distributions; log normal distribution, poisson distribution, binomial distribution
• sample statistic differ from pop
• sampling distribution centred around population mean

Question 5

standard error

Answer 5

standard deviation of sampling distribution
estimated from any sample
SE = SD/ square root of N
gauge accuracy of parameter estimate in sample
smaller SE, more likely parameter estimate is close to population parameter

Question 6

central limit theorem

Answer 6

• sampling distribution of mean is approximately normal, true no matter shape of population distribution
• as N gets larger, sampling distribution of sample mean tends towards normal distribution
• mean = µ, SD= SD/square root of N

Question 7

point estimates

Answer 7

• singel numbers that are best guesses about corresponding population parameters
• central tendency, measures of spread
• relationships between variables can be expressed using point estimates

Question 8

what does SE of mean express?

Answer 8

• uncertainty about relationship between sample and population mean
• sample mean is best estimate of population mean, true for all point estimates

Question 9

interval estimates

Answer 9

• communicate uncertainty around point estimate

- indicates how confident can be that estimate is representative of population parameter

Question 10

confidence interval (CI)

Answer 10

• using SE and sampling distribution to calculate CI with certain coverage
• 95% CI, 95% of intervals around sample estimate will contain value of population parameter
• 95% of sampl. distr. within +/- 1.96 SE, 95% CI estimate pop. mean is mean +/- SE

Question 11

t-distribution

Answer 11

• when don’t know sampling distribution
• symmetrical and centred around 0
• shape changes based on degrees of freedom
• ‘fat tailed’ when df=1; identical to normal dist. when df=infinite
• as df increases, tails get thinner
• critical value changes based on df
• df= N-1 (n is number of estimated parameters)

Question 12

what you need to calculate confidence intervals

Answer 12

estimated mean
sample SD
N
critical value fro t-distribution with df = N -1

-95% CI around estimated pop. mean is mean +/- SE

Question 13

CI’s are useful :

Answer 13

• width of interval tell us about how much we expect mean of different sample of same size to vary from one we got
• x% chance that any x% CI contains true population mean
• can be calculated for any point estimate

Question 14

hypothesis

Answer 14

• statement about something in terms of differences or relationships between things/people/groups
• must be testable
• about a single thing

Question 15

levels of hypotheses

Answer 15

• conceptual: expressed in normal language on level of concepts/constructs
• operational: restates conceptual hypothesis in terms of how constructs are measured in given study
• statistical: translates operational hypothesis into language of mathematics

Question 16

operationalisation

Answer 16

• process of defining variables in terms of how they are measured
• intelligence as total score on Ravens progressive matrics

Question 17

Statistical hypothesis

Answer 17

• operational hypothesis in terms of language of maths
• deals with specific values of population parameters
• mean of population can be hypothesised to be of given value
• can hypothesise a difference in means between two populations

Question 18

problems with samples that test hypothesis

Answer 18

not representative of population
larger the sample the better as fluctuations become less important as N increases
means converge to true value of population mean as N increases
CIs get exponentially smaller with N

Question 19

null hypothesis

Answer 19

states there is no difference

used to test for statistical significance

Question 20

distribution of test statistic under Ho

Answer 20

even if true difference in population delta is zero, D can be non-zero in sample
Assume A is normally distributed in population with µ=0 and o- = 1, expected value of D under Ho, more often than not D will not equal to 0 in sample

Question 21

what is a p-value

Answer 21

• the probability of getting test statistic at least as extreme as one observed if null hypothesis is true, how likely data is if there is no difference/effect in population
• if p-value is less than chosen significance level, call result statistically significant

Question 22

retain or reject null

Answer 22

reject null hypothesis when judge our result to be unlikely under Ho
retain Ho if judge result to be likely under it

Question 23

continuous data

Answer 23

• matter of degree eg how much
• score or measurement
• makes sense to have mean value

Question 24

categorical data

Answer 24

• matter of membership eg which group?
• group or label
• membership is binary

Question 25

for each statistical analysis we need:

Answer 25

data test statistic distribution of test statistic probability of value of test statistic uder null hypothesis

Question 26

correlation

Answer 26

- quantifies degree and direction of numeric relationship - used wtih two or more continuous variables or if one is categorical - use pearson correlation coefficient - only use correlated when reporting r as evidence

Question 27

what code in r is used to get pearsons correlation

Answer 27

data %>% select(variable, variable) %>% cor(method = 'pearson')

Question 28

what can you suggest when confidence intervals overlap

Answer 28

they may have same population value

Question 29

chi squared test

Answer 29

- quantifies relationship between two or more categorical variables - compare what might expect under null and calculate X^2 to quantify - only use X^2 if value greater than 5 in each cell - the bigger the X^2 value the bigger the difference between our data and what we expect

Question 30

important note about chi squared

Answer 30

only test significance of null hypothesis being true, there will be no evidence for alternative

Question 31

using t distribution

Answer 31

- t is the difference in sample means compared to standard error of differences in means - larger the t the bigger the difference bewteen sample means compared to error

Question 32

t and r

Answer 32

- p value from r comes from t distibution - can change t into r - t quantifies difference in means between two groups - R quantifies degree and direction of relationship between two variables

Question 33

what is a predicor

Answer 33

variable that may have relationship with outcome

Question 34

what is an outcome

Answer 34

variable we want to explain | outcome = model + error

Question 35

linear model

Answer 35

- creates linear model between outcome variable and predictor variable in dataset - look at lm() %>% summary() - R^2 is variance of variable A was explained by variable B - adjusted R^2 is if applied same model to population - R^2 and adjusted R^2 must be similar and big

Question 36

r code for linear model

Answer 36

Lm(outcome ~ predictor, data = data)

Question 37

equation of linear model

Answer 37

outcome = b0 + b1 x PREDICTOR1 + error

Question 38

f statistic

Answer 38

F = (what model can explain)/(what cant explain) - ratio of variance explained relative to variance unexplained - ratio > 1 means model can explain more than it cant - associated p value of how likely to find F stat as large as observed if null is true

Question 39

how to compare linear models

Answer 39

- compare R^2 and change in R^2 - compare f stat and its associated p-value - look at standardised versions of b1