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Statistics & Operations Research

STAT-UB 103

Ardeshir Shahmaei

Fall

Description

Standard error of the mean: standard deviation of the sampling distribution, e.g. √ or√
Margin of error, sampling error (SE): term added to/subtracted from ̅ when finding confidence
interval, or standard error multiplied by the half-width of confidence interval ( ⁄ , ⁄ ), e.g.
⁄ √ , ⁄ √ , or ⁄ √ depending on whether is known and value of .
Degrees of freedom: or
: level of significance
o Defines unlikely values of sample statistic if null hypothesis is true
Rejection regions: reject null hypothesis if observed test statistic is in these regions
o If test is target
Two tail test, where each tail has area
o If test is target
Upper tail test (right tail), where tail has area
o If test is target
Lower tail test (left tail), where tail has area
always has equals sign, never does
Error: actual population measurement does not match what was concluded with sample
measurements
o Type I ( ) error: rejected based on sample mean, when is true based on
population mean
o Type II ( ) error: is true based on sample mean, when rejected based on
population mean
Assumptions:
o Randomly selected
o Sample frequency distribution is normal, i.e.:
( )
̂ ̂
̂ ( ̂ ) ̂ ( ̂)
Given in the problem
Can find sample size by substituting in standard error in and rearranging the formula
o Difference of population variable of interest for two samples, used in . E.g. for mean:
( )
( )
( )
o Difference of ordered pairs
o ̅is mean of all differences in two dependent samples, is standard deviation of
all differences in two dependent samples
Hypothesis testing steps
o I. Specify the population variable of interest (e.g. , , ) o II. Formulate and by assigning population values of interest (e.g. )
o III. Specify
o IV. Draw rejection region based on . Label tail boundaries (e.g. , ⁄ )
o V. Compute the observed test statistic and see if it falls within tails (e.g. if , if
| | ⁄ )
o VI. Make decision to reject or not reject null hypothesis and restate hypothesis that was
supported (e.g. reject , )
-value is probability of observing sample result if null hypothesis is true
o Observed level of significance
o -value ( )
Let be the dead test statistic variable and be the observed test statistic
̅
value calculated from a formula (e.g. ⁄√ )
( ) ( ) | |
If two tail test: ( )
If upper tail test: ( )
If lower tail test: ( )
o If -value , reject
”Test for one sample”
o Hypothesis: ○
o If ( is given, or is given and can be used to )
̅
√ √
⁄ ̅
̅
Observed test statistic ̅
o If
̅ √
⁄ ̅
Observed test statistic ̅
̅
“Find confidence interval for ”
o ⁄
o [ ̅ ̅
“How large of a sample is needed for a given margin of error ( ⁄ widt

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