Hypothesis Tests About Hypothesis Tests About With With Unknown Unknown

Hypothesis Testing(Revisited)

• Five Step Procedure1. 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).

4. Take a random samplerandom 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:

X

nσ

value)zed(hypothesi - xz

ns

value)zed(hypothesi - xt

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:

5.43 29.5 s

29.5 118/4 4/)(-5)(8)(2)(0) ((-5) s

30 25)/5383230(25 x222222

EXAMPLE 1: “> TEST”• Is there enough evidence to conclude > 27?

1. H0: = 27

HA: > 27

2. = .05

3. Reject H0 (Accept HA) if t > t.05,4 = 2.132

4. Take Sample: 25, 30, 32, 38, 25

5. We get:

• There is not enough evidence to conclude > 27.

2.132is which 1.2355/43.5

)2730(t

EXAMPLE 2: “< TEST”• Is there enough evidence to conclude < 35?

1. H0: = 35

HA: < 35

2. = .10

3. Reject H0 (Accept HA) if t < -t.10,4 = -1.533

4. Take Sample: 25, 30, 32, 38, 25

5. We get:

• There is enough evidence to conclude < 35.

-1.533is which -2.0595/43.5

)3530(t

EXAMPLE 3: “ TEST”• Is there enough evidence to conclude 40?

1. H0: = 40

HA: 40

2. = .05

3. Reject H0 (Accept HA) if t > t.025,4 = 2.776

or if t < -t.025,4 = -2.776

4. Take Sample: 25, 30, 32, 38, 25

5. We get:

There is enough evidence to conclude 40.

2.776- is which 4.118- 5/43.5

)4030(t

EXCELt-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)

.n

s isentry second The

.x isentry first The

CHECK --Summary statistics

Confidence Level For Mean

EXCEL HYPOTHESIS TESTING “> TESTS”

• Refer to Example 1: HA: >27

• Calculate t by: =(Mean-27)/(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.

=(B3-27)/B4

=TDIST(E2,4,1)

1-tail

testDegrees of

freedomt

EXCEL HYPOTHESIS TESTING “< TESTS”

• Refer to Example 2: HA: <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)

-tbecause t < 0

Degrees of

freedom

1-tail

test

EXCEL HYPOTHESIS TESTING “ TESTS”

• Refer to Example 3: HA: 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