Stats 121 test 2

sampling distribution of the sample mean

a probability distribution of all possible sample means of a given sample size from the same population

central limit theorem

As the size n of a simple random sample increases, the shape of the sampling distribution of x̄ tends toward being normally distributed

sampling distribution center

population parameter, equivalent to mu of population

sampling distribution spread

standard deviation decreases as sample size increases

natural variation

common/normal sources of chance variation that does not cause problems

unnatural variation

special causes or assignable sources of variation
ex: bad batch of raw material, broken machine, poorly trained operator

statistical process control

a system in which management collects and analyzes information about the production process to pinpoint quality problems in the production system

x-bar control chart

statistical tool for monitoring a process that has variation, alerting us when a problem or unnatural variation has occurred (in which case process should be stopped and fixed)

in control process

process whose output exhibits only natural variation over time

out of control process

process exhibits unnatural variation

construction of x-bar control chart

1. draw horizontal centerline at μ
2. draw horizontal control limits at μ±3(σ/√3)
3. plot means from sample size n against time

out of control signals

point above/below control limits or nine consecutive points on the same side of the center line

inference

drawing conclusions about a population (parameter) based on data from a sample (statistic) with a measure of uncertainty

point estimation

data from the sample is used to estimate the population parameter; no measure of uncertainty; should only be used as step one in valid inference

interval estimation (confidence interval)

range of plausible values for population parameter; used for research questions asking for value

hypothesis testing (tests of significance)

states claim and checks whether sample data provides evidence for/against claim; used for research questions asking yes/no questions

accuracy of x-bar estimating μ

dependent on random sampling and distribution of x-bar (accuracy increases as sample size increases)

four steps for confidence intervals

state, plan, solve, conclude

confidence interval

estimate a parameter; value

test of significance

assess claim about a parameter; yes/no

conditions for inference

randomness, normal distribution, linear, no outliers, constant standard deviation

properties of t distributions

symmetric, bell shaped, mean=0, smaller degrees of freedom correlate to a larger spread, larger degrees of freedom correlates to be closer to the standard normal (z distribution)

format of t distribution table

- degrees of freedom = n-1
- use closest df without going over
- t* values found in the body of table

outline for one-sample t confidence interval

1. state the problem
2. plan (procedure, confidence level, parameter of interest in context)
3. solve (collect/plot data, calculate x-bar and s, check randomness and normality/large population, calculate)
4. conclude (state confidence, parameter in context and calculated interval)

confident

percentage of confidence intervals produced by the procedure that actually contain μ; success rate of procedure

margin of error

likely maximum difference between the statistic and the parameter at the stated confidence level; accounts for uncertainty due to sampling variability only

properties of a confidence interval

- margin of error (m) controls the width of the interval
- as sample size increases, m and width decrease (more precise)
- as sample size decreases, m and width increase (less precise)

when to use confidence intervals

randomness; normal population or large sample size

statistical inference

drawing conclusion about parameter using statistic with a measure of uncertainty

test of significance assumption

claim researchers think is not true; proof by contradiction

one sided test

a test with inequality in Ha

two sided test

a test with a not equal to in Ha

test statistic

a number that summarizes the data for a test of significance; compares estimate of parameter from sample data with parameter given in null hypothesis; measures how far sample data diverge from Ho; large values are not consistent with Ho and give evidence against; used to find probability of obtaining sample data if Ho were true

meaning of p-value

probability of getting a test statistic as extreme or more extreme than observed if Ho were true; measure of strength of agreement between observed test statistic and Ho (small = little agreement)

meaning of significance level (α)

pre-specified cutoff for p-value; boundary between rejection and non-rejection regions for p-value; if p-value is less than α, difference is statistically significant, reject Ho and conclude it's false

null hypothesis

always contains equality, claim we first assume is true and hope to disprove

alternative hypothesis

always contains inequality, claim we think is true and hope to prove by disproving Ho

p-value < α

statistically significant, reject Ho, sufficient evidence that Ha is true, difference between x-bar and claim is real

p-value > α

not statistically significant, fail to reject Ho, insufficient evidence that Ha is true, difference between x-bar and claim is due to chance

one-sample t-test for means

if SRS, unknown standard deviation, approximately normal population

then sampling distribution of equation has student's t-distribution with n-1 degrees of freedom

steps for one sample t-test

1. state problem (yes/no about quantitative)
2. plan (write Ho and Ha which both have mu, select α)
3. solve (compute test statistic and find p-value)
4. conclude (compare, fail/reject to fail, state sufficient/insufficient evidence in context)

standard error of x-bar

s/√n

margin of error for estimating mu

t*(s/√n)

p-value from t table

- df determine correct row
- follow columns on either side of test statistic and check for 1 or 2 sided test
- p-value is given as a range of values

significance depends on

size of observed effect (numerator), how far sample mean deviates from hypothesized claimed mean, size of sample

large observed effect and large sample size effects

smaller p-value

sample size and significance

sample size may be too small to detect significance or sample size may be so large results are always significant

practical importance

determined by common sense, not the same as statistical significance and is checked after, especially important for large samples

statistical significance

p-value < α

practically important

observed effect (numerator of test statistic) matters in real life

p-value for a two sided test

equal to two times the p-value for a one-sided test; requires stronger evidence (smaller probability) than one-sided test

confidence interval approach to hypothesis testing

Ha is two sided; confidence level and significance level add to 100%

confidence interval does not contain claimed mean

reject Ho, test is statistically significant

confidence interval contains claimed mean

fail to reject Ho, test is not statistically significant

type I error

rejecting Ho when Ho is true; false positive; probability: α

type II error

fail to reject Ho when Ho is false; false negative; probability: β

power

reject Ho when it's false; probability = 1-β

safe

fail to reject Ho when it's true; probability = 1-α

relationship between α and power (fixed n)

decreasing α increases β and decreases power; increasing α decreases β and increases power

relationship between n and power (fixed α)

increasing n increases power and decreases β

relationship between effect size and power (fixed α)

larger effect size results in larger power

effect size

difference between actual μ and claimed μ

small level of significance (α)

requires larger sample

higher power

requires larger sample

detecting a small effect size

requires a larger sample

a two sided test requires

a larger sample than a one sided test

α

intentionally set low

β

want low; done by increasing α or n

1-β

want high; done by increasing α or n or having a small spread or large effect size