Stats 121 test 2

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sampling distribution of the sample mean

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Statistics

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1

sampling distribution of the sample mean

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

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2

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

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3

sampling distribution center

population parameter, equivalent to mu of population

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4

sampling distribution spread

standard deviation decreases as sample size increases

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natural variation

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

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6

unnatural variation

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

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7

statistical process control

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

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8

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)

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9

in control process

process whose output exhibits only natural variation over time

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10

out of control process

process exhibits unnatural variation

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11

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

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12

out of control signals

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

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13

inference

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

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14

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

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15

interval estimation (confidence interval)

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

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16

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

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17

accuracy of x-bar estimating μ

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

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18

four steps for confidence intervals

state, plan, solve, conclude

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19

confidence interval

estimate a parameter; value

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20

test of significance

assess claim about a parameter; yes/no

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21

conditions for inference

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

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22

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)

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23

format of t distribution table

  • degrees of freedom = n-1

  • use closest df without going over

  • t* values found in the body of table

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24

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)

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25

confident

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

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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

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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)

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when to use confidence intervals

randomness; normal population or large sample size

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29

statistical inference

drawing conclusion about parameter using statistic with a measure of uncertainty

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30

test of significance assumption

claim researchers think is not true; proof by contradiction

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31

one sided test

a test with inequality in Ha

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32

two sided test

a test with a not equal to in Ha

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33

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

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34

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)

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35

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

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null hypothesis

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

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alternative hypothesis

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

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p-value < α

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

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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

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40

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

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41

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)

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42

standard error of x-bar

s/√n

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43

margin of error for estimating mu

t*(s/√n)

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44

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

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45

significance depends on

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

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46

large observed effect and large sample size effects

smaller p-value

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47

sample size and significance

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

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48

practical importance

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

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49

statistical significance

p-value < α

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50

practically important

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

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51

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

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52

confidence interval approach to hypothesis testing

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

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53

confidence interval does not contain claimed mean

reject Ho, test is statistically significant

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54

confidence interval contains claimed mean

fail to reject Ho, test is not statistically significant

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55

type I error

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

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type II error

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

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57

power

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

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safe

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

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relationship between α and power (fixed n)

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

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60

relationship between n and power (fixed α)

increasing n increases power and decreases β

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61

relationship between effect size and power (fixed α)

larger effect size results in larger power

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62

effect size

difference between actual μ and claimed μ

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63

small level of significance (α)

requires larger sample

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64

higher power

requires larger sample

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65

detecting a small effect size

requires a larger sample

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66

a two sided test requires

a larger sample than a one sided test

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67

α

intentionally set low

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68

β

want low; done by increasing α or n

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69

1-β

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

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