Hypothesis Tests About  With  Unknown

Hypothesis Testing (Revisited) Five Step Procedure 1.Define Opposing Hypotheses. (  ) 2.Choose a level of risk (  ) for making the mistake of concluding something is true when its not. 3.Set up test (Define Rejection Region). random sample 4.Take a random sample. 5.Calculate statistics and draw a conclusion.

t Statistic in Hypothesis Tests for - (  UNKNOWN) When  is known we used: When  is unknown we use:

EXAMPLE Assuming that the ages of MIS managers follow a normal distribution, suppose we wish to draw conclusions about their true mean age (using α =.05) given the following random sample of ages of 5 MIS managers: 25, 30, 32, 38, 25. For this sample:

EXAMPLE 1: “> TEST” Is there enough evidence to conclude  > 27? 1.H 0 :  = 27 H A :  >  = Reject H 0 (Accept H A ) if t > t.05,4 = Take Sample: 25, 30, 32, 38, We get: There is not enough evidence to conclude  > 27.

EXAMPLE 2: “< TEST” Is there enough evidence to conclude  < 35? 1.H 0 :  = 35 H A :  <  = Reject H 0 (Accept H A ) if t < -t.10,4 = Take Sample: 25, 30, 32, 38, We get: There is enough evidence to conclude  < 35.

EXAMPLE 3: “  TEST” Is there enough evidence to conclude   40? 1.H 0 :  = 40 H A :    = Reject H 0 (Accept H A ) if t > t.025,4 = or if t < -t.025,4 = Take Sample: 25, 30, 32, 38, We get: There is enough evidence to conclude   40.

EXCEL t-TESTS For all hypothesis tests, first get the mean and the standard error (s/  n) as follows: Go to DESCRIPTIVE STATISTICS -- Check –Summary Statistics –Confidence Level for Mean (indicate % confidence)

CHECK -- Summary statistics Confidence Level For Mean

EXCEL HYPOTHESIS TESTING “> TESTS” Refer to Example 1: H A :  >27 Calculate t by: =(Mean-27)/(Standard Error) p-value: if t >0, =TDIST(t,4,1) gives a p <.5 if t.5 Numbers in italics means click on the cell with this value.

=(B3-27)/B4 =TDIST(E2,4,1) 1-tail test Degrees of freedom t

EXCEL HYPOTHESIS TESTING “< TESTS” Refer to Example 2: H A :  <35 Calculate t by: =(Mean-35)/(Standard Error) p-value: if t <0, =TDIST(-t,4,1) gives a p <.5 if t >0, =1-TDIST(t,4,1) gives a p >.5 Numbers in italics means click on the cell with this value.

=(B3-35)/B4 =TDIST(-E2,4,1) -t because t < 0 Degrees of freedom 1-tail test

EXCEL HYPOTHESIS TESTING “  TESTS” Refer to Example 3: H A :   40 Calculate t by: =(Mean-40)/(Standard Error) p-value:=TDIST(ABS(t),4,2) Numbers in italics means click on the cell with this value.

=(B3-40)/B4 =TDIST(ABS(E2),4,2) To make sure the first argument is >0 Degrees of freedom 2-tail test

REVIEW t-tests the same as z-tests except: –use s instead of  –use t instead of z Excel –Use Descriptive Statistics to get sample mean and standard error –Use of TDIST function