Stats 121 test 2

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

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

population parameter, equivalent to mu of population

standard deviation decreases as sample size increases

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

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

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

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)

process whose output exhibits only natural variation over time

process exhibits unnatural variation

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

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

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

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

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

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

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

state, plan, solve, conclude

estimate a parameter; value

assess claim about a parameter; yes/no

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

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)

• degrees of freedom = n-1

• use closest df without going over

• t* values found in the body of table

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)

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

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

• 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)

randomness; normal population or large sample size

drawing conclusion about parameter using statistic with a measure of uncertainty

claim researchers think is not true; proof by contradiction

a test with inequality in Ha

a test with a not equal to in Ha

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

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)

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

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

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

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

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

if SRS, unknown standard deviation, approximately normal population

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

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)

s/√n

t*(s/√n)

• 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

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

smaller p-value

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

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

p-value < α

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

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

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

reject Ho, test is statistically significant

fail to reject Ho, test is not statistically significant

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

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

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

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

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

increasing n increases power and decreases β

larger effect size results in larger power

difference between actual μ and claimed μ

requires larger sample

requires larger sample

requires a larger sample

a larger sample than a one sided test

intentionally set low

want low; done by increasing α or n

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