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Hypothesis Testing For Hypothesis Testing For With With Known Known
HYPOTHESIS TESTING
• Basic idea: You want to see whether or not your data supports a statement about a parameter of the population.– The statement might be: The average age of night
students is greater than 25 ( >25)
• To do this you take a sample and compute x
• If < 25 – You did not prove >25.
• If >>25– You are probably satisfied that >25.
• If is slightly > 25– You are probably not convinced that >25.
(although there is some evidence to support this)
x
Results x
x
x
Being Convinced• Hypothesis testing is like serving on a jury.serving on a jury.
– The prosecution is presenting evidence (the data) to show that a defendant is guilty (a hypothesis is true).
– But even though the evidence may indicate that the defendant might be guilty (the data may indicate that the hypothesis might be true), you must be convinced beyond a “reasonable doubt”.“reasonable doubt”.
• Otherwise you find the defendant “not guilty”– this does not mean he was innocent, (there is not enough evidence to support the hypothesis – this does not mean that the hypothesis is not true, just that there was not enough evidence to say it was true beyond a reasonable doubt).
When Can You Conclude > 25?
• You are convinced >25 if you get an that is “a lot greater”“a lot greater” than 25.
• How much is “a lot”“a lot” ?– This is hypothesis testing.
x
25 above is valuexyour ),X of deviations standard -- n
(
errors standardmany howin liesanswer The
aboveabove
“<” Tests
• Can you conclude that the average age of night students is less than 27? ( < 27)
27 below is valuexyour ),X of deviations standard -- n
(
errors standardmany howin liesanswer The
belowbelow
“” Tests
• Can we conclude that the average age of students is different from 26? ( 26)
26 below)or (above fromaway is value
xyour ),X of deviations standard -- n
(
errors standardmany howin liesanswer The
away fromaway from
Hypothesis Testing
• 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.
STEP 1: Defining Hypotheses
• H0 (null hypothesis -- status quo)
• HA (alternate hypothesis -- what you are trying to show)
Can we conclude that the Can we conclude that the average age exceeds 25?average age exceeds 25?
H0: 25
HA: > 25
Test H0 “At the break point” = 25• Test the hypothesis at the point that would give
us the most problem in deciding if >25. – If really were 1515, it would be very unlikelyvery unlikely that we would
draw a sample of students that would lead us to the falsefalse conclusion that >25.
– If really were 2222, it is more likelymore likely that we would draw a sample with a large enough sample mean to lead us to the falsefalse conclusion that >25; if really were 2424, it would be even more likelymore likely, 24.924.9 even more likely.
= 25= 25 is the “most likely” value in H0 of generating a sample mean that would lead us to the falsefalse conclusion that >25.
So H0 is tested “at the breakpoint”, = 25
STEP 2: Choosing a Measure of Risk(Selecting )
= P(concluding HA is true when it is not)
• Typical values are .10, .05, .01, but can be anything.– Values of are often specified by professional
organizations (e.g. audit sampling normally uses values of .05 or .10).
STEP 3: How to Set Up the TestDefining the “Rejection Region”
• Depends on whether HA is a >, <, or hypothesis.
• The “> Case”– Suppose we are hypothesizing > 25. > 25.– When a random sample of size n is taken:
• If really = 25, then there is only a probability = that we would get an value that is more than z standard errors above 25.
• Thus if we get an value that is greater thangreater than z standard errors above 25, we are willing to conclude > 25.
– We call this critical value xxcritcrit.
x
x
n
σz 25 x αcrit
Rejecting H0 (Accepting HA)
00 ZZX
n
σσX
2525
=25if H0 is true
REJECTIONREGION
critx
Reject H0 (Accept HA) if we get
)z(z xx αcrit
zz
How many standard errors away is ?
• This is called the test statistictest statistic, z
x
error standard eappropriat
value)zed(hypothesi - estimate)point (z
nσ
25 - xz
A ONE-TAILED “>” TEST
• Reject H0 (Accept HA) if:
z > z
• Values of z
z
.10 1.282
.05 1.645
.01 2.326
A ONE-TAILED “<“ TEST
• If HA were, HA: < 27• The test would be: Reject H0 (Accept HA) if:
z < -z
• Values of -z
-z
.10 -1.282
.05 -1.645
.01 -2.326
A TWO-TAILED “” TEST
• If HA were, HA: 26
• The test would be: Reject H0 (Accept HA) if:
z < -z/2 or z > z/2
• Values of -z
-z/2 z/2
.10 -1.645 1.645
.05 -1.96 1.96
.01 -2.576 2.576
STEP 4: TAKE SAMPLE
• After designing the test, we would take the sample according to a randomrandom sampling procedure.
STEP 5: CALCULATE STATISTISTICS
• From the sample we would calculate
• Then calculate:
x
nσ
25 - xz
DRAWING CONCLUSIONSOne Tail Tests
• ““>” TEST>” TEST: HHAA: : > 25 > 25
If z > zz > z -- Conclude HA is true (Reject H(Reject H00))
If z < zz < z -- Cannot conclude HA is true
(Do not reject H(Do not reject H00))
• ““<” TEST:<” TEST: HHAA: : < 27 < 27
If z < -zz < -z -- Conclude HA is true (Reject H0)
If z > -zz > -z -- Cannot conclude HA is true
(Do not reject H(Do not reject H00))
DRAWING CONCLUSIONSTwo Tail Tests
HHAA: : 26 26
• Case 1 -- When z > 0Case 1 -- When z > 0: compare z to zz to z/2/2
If z > zz > z/2/2 -- Conclude HA is true (Reject H0)
If z < zz < z/2/2 -- Cannot conclude HA is true (Do not reject H(Do not reject H00))
• Case 2 -- When z < 0Case 2 -- When z < 0: compare z to -z/2
If z < -zz < -z/2/2 -- Conclude HA is true (Reject H0)
If z > -zz > -z/2/2 -- Cannot conclude HA is true (Do not reject H(Do not reject H00))
DRAWING CONCLUSIONSTwo Tail Tests (Alternative)
HHAA: : 26 26
• Cases 1 and 2 can be combined by simply looking at |z|. The test becomes:
Compare |z| to zCompare |z| to zαα/2/2
If |z| > z|z| > z/2/2 -- Conclude HA is true (Reject H0)
If |z|< z |z|< z/2/2 -- Cannot conclude HA is true
(Do not reject H(Do not reject H00))
Examples
• Suppose– We know from long experience that = 4.2 – We take a sample of n = 49 students– We are willing to take an = .05 chance of concluding
that HA is true when it is not (Note: z.05 = 1.645)
• Because our sample is large, a normal distribution approximates the distribution of X
Example 1: Can we conclude > 25?
1. H0: = 25
HA: > 25
2. = .05
3. Reject H0 (Accept HA) if:
4. Take sample 25,21,… 33.
1.645 z
492.4
25 - x z .05
25. age average that theconclude toevidenceenough is There 1.645. 2.05
05.2
494.2
25)-(26.23 z 26.23x .5
Example 2: Can we conclude < 27?1. H0: = 27
HA: < 27
2. = .05
3. Reject H0 (Accept HA) if:
4. Take sample 22,28,… 33.
1.645- z
492.4
27 - x z .05
.)27μ THAT CONCLUDE NOT DO (WE 27. age average hat the t
conclude toevidenceenough not is There 1.645. - not is 1.2833-
2833.1
494.2
27)-(26.23 z 26.23x .5
NOTE!
Example 3: Can we conclude 26?1. H0: = 26
HA: 26 (This is a two-tail test)
2. = .05
3. Reject H0 (Accept HA) if:
4. Take sample 25,21,… 33.
1.96 z
492.4
26 - x |z| .025
)26.μ THAT CONCLUDE NOT DO (WE 26. from differs age average
that theconclude toevidenceenough not is There .96.1not is .383
383.
494.2
26)-(26.23 |z| 26.23x .5
NOTE!
> TESTS> TESTS
Given Values
2.05102 > 1.6448532.05102 > 1.644853
Can conclude mu > 25Can conclude mu > 25
=NORMSINV(1-C2)
=AVERAGE(A2:A50)
(C7-C3)/(C1/SQRT(49))
< TESTS< TESTS
-1.28231 > -1.64485-1.28231 > -1.64485
Cannot conclude mu < 27Cannot conclude mu < 27
Given Values
=-NORMSINV(1-C2)
=AVERAGE(A2:A50)
(C14-C10)/(C1/SQRT(49))
TESTSTESTS
Given Values
.384354 < 1.959963.384354 < 1.959963
Cannot conclude mu Cannot conclude mu 26 26
=ABS((C21-C17)/(C1/SQRT(49)))
=NORMSINV(1-C2/2)
=AVERAGE(A2:A50)
REVIEW• “Common sense concept” of hypothesis testing• 5 Step Approach
– 1. Define H0 (the status quo), and HA (what you are trying to show.)
– 2. Choose = Probability of concluding HA is true when its not.
– 3. Define the rejection region and how to calculate the test statistic.– 4. Take a random sample.– 5. Calculate the required statistics and draw conclusion.
• There is enough evidence to conclude HA is true (Reject H0)
• There is not enough evidence to conclude HA is true (Do not reject H0).