Unit 8 - Statistical Inference

probability

the likelihood that any one event will occur, given all the possible outcomes

what is probability symbol

p = probability (ratio or decimal)

distribution of scores

the general shape of data which includes a mean, median, and mode

example of distribution of scores

height of adults

- mean = 69 inches (z score = 0)

- standard deviation= +- 3 inches

standard deviation

show how spread out the scores are.

e.g +-3 (3 sd away from the mean)

where is probability under distribution curve?

the more area there is under the graph for a certain score equals probability of those scores

sampling distribution of means

A frequency distribution showing all possible sample means that occur when samples of a particular size are drawn from a population

assumption of sampling distribution

samples are randomly selected and valid representations of the population

sampling error

the larger the error, the less accurate the sample represents the population

- as sample size increases, variability from mean is reduced

type of inferential statistics that does not establish cause and effect

relation/prediction

- correlation & regression analysis

type of inferential statistics that tries to establish cause & effect

1 group -> one sample z-test or t-test

2 groups or 1 group measured twice -> 2 sample t-test or paired t-test

more than 2 groups -> ANOVA

what if sample size is small

- use t-distribution, instead of z-distribution as variability decreases with larger sample size

- t-distrbution curves are more platykurtic

- t-distribution also depends on sample size

hypothesis

a declarative statement that predicts the relationship between the independent and dependent variables, specifying the population that will be studied

two types of hypothesis

- research hypothesis

- null hypothesis

research hypothesis

states the researcher's true expectations of results guiding the interpretation of outcomes and conclusions

- depends on the field of study

null hypothesis

statistical hypothesis where statistically testing of data is allowed once data is collected as part of the research study

- does not depend on the field of study

criteria for hypothesis testing

characteristics and inclusion criteria for participants to be grouped must be specific to the population to which conclusion are to be inferred to

how is statistical null hypothesis stated

objective of null hypothesis

to be able to use statistic to infer back to respective population (why population symbol is used)

how is statistical alternative hypothesis stated

directional hypothesis

researchers have reasonable expectations that one group is greater or less than another

- directional doesn't play out in null hypothesis, only alternative hypothesis

errors in hypothesis testing

-must ONLY accept or reject null hypothesis

- when accepting null hypothesis, you reject alternative hypothesis (vise versa)

- based on results of statistical test (i.e, calculations)

Type I error (alpha)

False positive results

ex: reject the null hypothesis when you should accept it

at what % of error do researcher usually stays at?

5% (1% is when researchers are very confident they didn't make mistake)

Type II error (beta)

false negative

ex: accept null hypothesis when you should reject it

statistical power

the probability that statistical tests will lead to rejection of the null hypothesis (ex: when ejecting null, you are stating 2 or more groups are significantly different)

factors affecting statistical power

1) alpha level chosen

- selecting 5% vs 1% risk of type i error will increase statistical power

- using one-tailed vs two tailed approach will increase statistical power

2) sample size

- greater sample size = tighter distribution

- smaller sample size = wider distribution

3) variability

- large sigma: wider distribution of scores which will decrease statistical data

- smaller sigma: tight distribution will increase statistical data

4) the magnitude of differences between groups

- greater differences between means = larger statistical power

- magnitude is determined by effect size

statistical testing

using established statistical testing (equations) to determine the acceptance or rejection of a null hypothesis

z-score

difference between raw score and the group (sample) mean divided by the standard deviation

- it's unitless

one sample z-test

determine if sample is representative of population

critical region

the area in the tails of the comparison distribution in which the null hypothesis can be rejected

noncritical region

the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected

steps in hypothesis testing

1. state null hypothesis in symbols and words

2. state alternative hypothesis in symbols and words

3. choose an alpha level (usually 5%) and one or two-tailed (often used where 5% is split between upper & lower tail)

4. state rejection and retain rule

5. compute appropriate statistic (z test)

6. make decision by applying rejection/retain rule

7. write conclusion in context of study