# MBS Chapters 5-7, parts of chapter 4

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School
Department
Statistics & Operations Research
Course
STAT-UB 103
Professor
Ardeshir Shahmaei
Semester
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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