Studied by 3 people
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
draw horizontal centerline at μ
draw horizontal control limits at μ±3(σ/√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
state the problem
plan (procedure, confidence level, parameter of interest in context)
solve (collect/plot data, calculate x-bar and s, check randomness and normality/large population, calculate)
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
state problem (yes/no about quantitative)
plan (write Ho and Ha which both have mu, select α)
solve (compute test statistic and find p-value)
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
Note 
Flashcard (53)