stats exam two

sampling error

error that exists between sample statistic and population parameter. measure of the average distance between each sample mean and population mean

distribution of sample means 

collection of sample means for all possible samples of size from a population 

sample means pile up around population means

pile of sample means forms a relatively normal distribution 

the larger the sample size, the closer the sample mean is the population mean

distribution of sample means characteristics

central limit theorem 

for any population with mean and standard deviation, the distribution of sample means for sample size will have a mean and standard deviation and will approach a normal distribution as the sample size approaches infinity (30+)

law of large numbers (standard error)

the larger the sample size, the more likely it is that the sample mean is close to the population mean

standard error formula

sigma / square root of n

z score distribution of sample means formula 

sample mean - mean of distribution/ standard error (sigma underscore M)

hypothesis testing

statistical test that uses sample data to evaluate a hypothesis about the population

hypothesis testing steps (z score)

1. state the hypothesis

2. set the criteria for a decision

3. compute z-score

4. make a decision

null hypothesis 

predicts no change, difference, or relationship. the IV has no effect on the DV

alternative hypothesis

predicts a change, difference, or relationship. the IV has an effect on the DV

alpha level (level of significance)

probability value used to define the concept of low probability in a hypothesis test

critical region 

composed of the extreme, low probability sample values. end of the tail 

type 1 error

incorrectly rejecting the null hypothesis. concluding that treatment does have an effect when it really does not

type 2 error

incorrectly retaining the null hypothesis. concluding that treatment doesn’t have an effect when it really does

directional hypothesis test (one tailed test)

hypothesis specify either an increase or decrease in the population mean. makes statement about the direction of effect

statistical signigficance means that the

Ho has been rejected

random sampling

randomized sample from a population. participants must not have a relationship

independent observations

No consistent, predictable relationships between 2+ observations

hypothesis test assumptions 

sigma is not changed by treatment, variable is normally distributed 

issues with hypothesis testing

focuses on data rather than hypothesis, data suggests that this sample mean is very unlikely (p < .05) if the null hypothesis is true

significant effect is not equal to substaintal effect

effect size 

measures the size of treatment effect. % of the variability in the DV can be attributed to the IV 

cohen’s d

measure of effect size

cohen’s d formula

mean difference/ standard difference

power

probability that the test will correctly reject a false null hypothesis and identify a treatment effect if it exists 

power is influenced by

effect size, sample size, alpha size, one tailed vs two tailed

power influence- effect size

as effect size increases, power increases

power influence-sample size

both increase

power influence- alpha level 

both decrease

power influence- one tail vs two tail

one tailed test as more power then two tailed test

functions of alpha 

defines very unlikely outcomes

determines probability of type 1 errors 

smaller alpha makes it harder to reject Ho

issue with z-scores

don’t usually know the population standard deviation

when do you use z-scores

hypothesis tests when sigma is known

when do you use t-scores

hypothesis test when sigma is unknown

t-scores

used to test hypothesis when population standard deviation is unknown

t-score formula

sample mean (M)- population mean (mu)/ estimated standard error (Sm)

estimated standard error

estimates the true standard error when sigma is unknown 

calculating t-score steps

1. calculate s

2. calculate Sm

3. calculate t-score

degrees of freedom

number of scores in a sample that are free to vary

t-distribution

complete set of t-scores computed for every possible random sample for a specific sample size 

t-test steps 

1. state hypothesis 

2. set criteria for decision 

3. compute t-score 

4. make a decision 

estimated d

estimate of effect size when sigma is unknown

t score formula for s

square root of ss/n-1

t score formula for sample mean 

s/square root of n 

t score formula for t

sample mean- population mean / sample mean answer