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tutorial07 - ISMT 111 L1 BUSINESS STATISTICS TUTORIAL 7...

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1 ISMT 111 L1 B USINESS S TATISTICS T UTORIAL 7 created by Andrew Yam Definition § Null Hypothesis ( 0 H ) - a hypothesis we assume to be TRUE, e.g. 0 0 : m m = H - assumption we wish to test Alternative Hypothesis ( a H or 1 H ) - a hypothesis we wish to support, e.g. 0 : m m a H - conclusion when we reject 0 H Observed Test Statistics ( obs z or obs t ) - a value used to make the decision, e.g. n X z obs / 0 s m - = Critical Value ( c ) - a value used to separate the rejection and non-rejection regions a - the Level of Significance b - 1 - the Power of the Test p -value - the probability that the test statistic would take a value as or more contradictory to 0 H than the observed value - the smallest value of a to reject 0 H , e.g. ( 29 obs z z P Type I and Type II Error § Type I error: reject 0 H when it is TRUE. P(committing Type I error) = P(reject 0 H | 0 H is true) = a § Type II error: do not reject 0 H when it is FALSE. P(committing Type II error)
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2 0 H is TRUE 0 H is FALSE Reject 0 H Type I error a (Significance Level) ü b - 1 (Power of the Test) Do NOT Reject 0 H ü a - 1 (Confidence Coefficient) Type II error b § Note: - For fixed n , ↑⇒ b a and ↓⇒ b a - ↑⇒ b a and n Hypothesis Testing of a Population Mean, m § Null Hypothesis, 0 0 : m m = H § z -test (Large Sample) - Normal Population; s is known - OR Non-normal Population; unknown s ; but large sample size n Test Statistics: n X z obs / 0 s m - = i) 0 : m m < a H , 0 H is rejected when a z z obs - < , ii) 0 : m m a H , 0 H is rejected when a z z obs , iii) 0 : m m a H , 0 H is rejected when 2 / a z z obs - < or 2 / a z z obs . Note: - When n is large, unknown s can be replaced by s . - Cases i) and ii) are called One-Tailed Test, while case iii) is called Two-Tailed Test. § t -test (Small Sample)
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3 - Assumption: Normal Population with unknown s i) 0 : m m < a H , 0 H is rejected when a t t obs - < , ii) 0 : m m a H , 0 H is rejected when a t t obs , iii) 0 : m m a H , 0 H is rejected when 2 / a t t obs - < or 2 / a t t obs , where 1 d.f. - = n and n s X t obs / 0 m - = . § p -value - p -value a < Reject 0 H at a level - p -value a Do not reject 0 H at a level - For z- test, exact p -value can be obtained from the normal table. Case i): p -value ( 29 obs z Z P < = Case ii): p -value ( 29 obs z Z P = Case iii): p -value ( 29 ( 29 | | 2 | z | or Z | | obs obs obs z Z P z Z P × = - < = - For t- test, a range of p -value can be obtained from the t - table (not the exact p -value) Hypothesis Testing of the Difference between Two Population Means µ 1 - µ 2 § Assumption: Two populations are independent. § Null Hypothesis, 0 2 1 0 : D H = - m m § z -test (Large Samples) - Normal Populations; 1 s and 2 s are known - OR Non-normal Population; unknown i s ; but BOTH 1 n and 2 n are large
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4 Test Statistics: ( 29 2 2 2 1 2 1 0 2 1 n n D X X z obs s s + - - = i) 0 2 1 : D H a < - m m , 0 H is rejected when a z z obs - < , ii) 0 2 1 : D H a - m m , 0 H
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This note was uploaded on 09/15/2008 for the course ISMT 111 taught by Professor Wanxuhu during the Fall '08 term at HKUST.

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tutorial07 - ISMT 111 L1 BUSINESS STATISTICS TUTORIAL 7...

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