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7/28/2019 Hypothesis of Large Scale
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7/28/2019 Hypothesis of Large Scale
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Piyoosh Bajoria 2
• Contents.
• 1. Elements of a statistical test
• 2. A Large-sample statistical test• 3. Testing a population mean
• 4. Testing a population proportion
• 5. Testing the difference between two populationmeans
• 6. Testing the difference between two population
proportions
• 7. Reporting results of statistical tests: p-Value
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Varsha Varde 3
Mechanics of Hypothesis Testing
• Null Hypothesis :Ho:
What You Believe(Claim/Status quo)
• Alternative Hypothesis:Ha: The Opposite ( proveor disprove with samplestudy)
3Piyoosh Bajoria
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Ha is less than type or left-tail test• 1. One-Sided Test of Hypothesis:
• < (Ha is less than type or left-tail test).• To see if a minimum standard is met
• Examples
• Contents of cold drink in a bottle
• Weight of rice in a pack
• Null hypothesis (H 0 ) : : µ = µ
0 Alternative hypothesis (Ha): : µ < µ0
Piyoosh Bajoria 4
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Ha is more than type or right -tail test
• One-Sided Test of Hypothesis:
• > (Ha is more than type or right -tail test).
• To see that maximum standards are not
exceeded.
• Examples
• Defectives In a Lot
• Accountant Claims that Hardly 1%
Account Statements Contain Error
• . Null hypothesis (H 0 ): p = p0
Alternative hypothesis (Ha): p > p0
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Two-Sided Test of Hypothesis: • Two-Sided Test of Hypothesis:
• ≠ (Ha not equal to type)• Divergence in either direction is critical
• Examples
• Shirt Size of 42
• Size of Bolt & nuts
• Null hypothesis (H 0 ) : µ = µ0
Alternative hypothesis (Ha): µ ≠ µ0
Piyoosh Bajoria 6
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DEFINITIONS
• Type I error ≡{ reject H 0 |H 0 is true }
• Type II error ≡{ do not reject H 0 |H 0 isfalse}
• α = Prob{Type I error}• β= Prob{Type II error}
• Power of a statistical test:
Prob{reject H 0 |H 0 is false }= 1- β
Varsha Varde 77Piyoosh Bajoria
EXAMPLE
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EXAMPLE• Example 1.
• H 0 : Innocent
• Ha: Guilty
• α = Prob{sending an innocent person to jail}
• β= Prob{letting a guilty person go free}
• Example 2.
• H 0 : New drug is not acceptable
• Ha: New drug is acceptable
• α = Prob{marketing a bad drug}• β= Prob{not marketing an acceptable drug}
Piyoosh Bajoria 8
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GENERAL PROCEDURE FOR HYPOTHESIS TESTING
• Formulate the null & alternative hypothesis
• Equality Sign Should Always Be In NullHypothesis
• Choose the appropriate sampling distribution
• Select the level of significance and hence thecritical values which specify the rejection and
acceptance region
• Compute the test statistics and compare it to
critical values
• Reject the Null Hypothesis if test statistics falls in
the rejection region .Otherwise accept it
9Piyoosh Bajoria
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• Null hypothesis: H 0
• Alternative (research) hypothesis: H a• Test statistic:
• Rejection region : reject H 0 if .....
• Decision: either “Reject H 0 ” or “Do not reject H 0 ”
• Conclusion: At 100α % significance level there is
(in)sufficient statistical evidence to “ favour Ha” . • Comments:
• * H 0 represents the status-quo
• * Ha is the hypothesis that we want to provideevidence to justify. We show that Ha is true by
showing that H 0 is false, that is proof by contradiction.
1010
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A general Large-Sample Statistical Test
• Parameter of interest: θ
• Sample data: n, ˆθ, σ ̂θ
• Other information: µ0 = target value,
α = Level of significance
• Test:Null hypothesis (H 0 ) : θ = θ 0
: Alternative hypothesis (Ha):
1) θ > θ 0 or
2) θ <θ 0 or
3) θ ≠θ 0
• Test statistic (TS): z =( ̂θ - θ 0 )/ σ ̂θ
• Critical value: either z α or z α /2
Piyoosh Bajoria 11
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A General Large-Sample Statistical Test
• Rejection region (RR) :
• 1) Reject H0 if z > z α • 2) Reject H0 if z < - z α
• 3) Reject H0 if z > z α /2 or z < -z α /2
Decision: 1) if observed value is in RR: “Reject H0”
• 2) if observed value is not in RR: “Do no reject H0”
• Conclusion: At 100α% significance level thereis (in)sufficient statistical evidence to…….. .
• Assumptions: Large sample + others (to be
specified in each case).
Piyoosh Bajoria 12
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T ti P l ti M
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Testing a Population Mean
• Conclusion: At 100α%
significance level there is(in)suficient statistical evidence to
“ favour Ha” . • Assumptions:
• Large sample (n ≥30)
• Sample is randomly selected
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EXAMPLE• Example: It is claimed that weight loss in a new diet
program is at least 20 pounds during the first month.
Formulate &Test the appropriate hypothesis• Sample data : n = 36, x¯ = 21, s2 = 25, µ0 = 20, α = 0.05
• H0 : µ ≥20 ( µ is 20 or larger)
• Ha : µ < 20 (µ is less than 20)
• T.S. :z =(x - µ0 )/(s/√n)=21 – 20/ 5 /√36 = 1.2
• Critical value: z α = -1.645
• RR: Reject H0 if z < -1.645
• Decision: Do not reject H0 • Conclusion: At 5% significance level there is insufficient
statistical evidence to conclude that weight loss in a new
diet program exceeds 20 pounds per first month.
Pi oosh Ba oria
15
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Testing a Population Proportion
• Parameter of interest: p (unknown parameter)
• Sample data: n and x (or p = x/n)
• p0 = target value• α (significance level)
• Test:H0 : p = p0
• Ha: 1) p > p0 ; 2) p < p0 ; 3) p = p0
• T.S. :z =( p - p0 )/√p0 q0 /n
• Rejection region (RR) :
• 1) Reject H0 if z > z α
• 2) Reject H0 if z < - z α • 3) Reject H0 if z > z α /2 or z < -z α /2
• Decision: 1) if observed value is in RR: “Reject H0”
• 2) if observed value is not in RR: “Do no reject H0”
• Assumptions:1. Large sample (np≥ 5, nq≥ 5) 2. Sample isPiyoosh Bajoria
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Example• Test the hypothesis that p > .10 for sample data:
• n = 200, x = 26.
• Solution. p = x/n = 26/ 200 = .13,
• H0 : p = .10 (p is not larger than .10)
• Ha : p > .10
• TS:z = (p - p0 )/√p0 q0 /n=.13 - .10/√(.10)(.90)/200 = 1.41• RR: reject H0 if z > 1.645
• Dec: Do not reject H 0
• Conclusion: At 5% significance level there is insufficient
statistical evidence to conclude that p > .10.• Exercise: Is the large sample assumption satisfied
here ?
Piyoosh Bajoria17
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Comparing Two Population Means
• Parameter of interest: µ1 - µ2
• Sample data:
• Sample 1: n1, x¯ 1, s1
• Sample 2: n2 , x¯ 2 , s2
• Test:
• H 0 : µ1 - µ2 = D0
• Ha : 1)µ1 - µ2 > D0 ; 2) 1)µ1 - µ2 < D0 ;3) µ1 - µ2 = D0
• T.S. :z =( x¯ 1 - x¯ 2 ) - D0 /√σ 2 1/n1+ σ2
2/n2
• RR:1) Reject H0 if z > z α ;2) Reject H0 if z < -z α
• 3) Reject H0 if z > z α /2 or z < -z α /2
• Assumptions:• 1. Large samples ( n1≥ 30; n2 ≥30)
• 2. Samples are randomly selected
• 3. Samples are independent
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• Refer to the weight loss example. Test the hypothesis
that weight loss in the two diet programs are different.
• 1. Sample 1 : n1 = 36, x¯ 1 = 21, s2 1 = 25 (old)
• 2. Sample 2 : n2 = 36, x¯ 2 = 18.5, s2 2 = 24 (new)
• D0 = 0, α = 0.05
• H0 : µ1 - µ2 = 0
• Ha : µ1
- µ2
≠ 0,
• T.S. :z =( x¯ 1 - x¯ 2 ) – 0/√σ 2 1/n1+ ó2
2/n2 = 2.14
• Critical value: z α /2 = 1.96
• RR: Reject H0 if z > 1.96 or z < -1.96
• Decision: Reject H0 • Conclusion: At 5% significance level there is sufficient
statistical evidence to conclude that weight loss in the
two diet programs are different.
Piyoosh Bajoria 19
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• Parameter of interest: p1 - p2
• Sample 1: n1, x 1, ˆp1 = x 1 /n1
• Sample 2: n2 , x 2 , ˆp2 = x 2 /n2
• p1 - p2 (unknown parameter)• Common estimate: ̂p =(x1 + x2)/(n1 + n2)
• Test:H 0 : p1 - p2 = 0
• Ha : 1) p1 - p2 > 0;2) p1 - p2 < 0;3) p1 - p2 = 0
• TEST STATISTICS:z =( ̂p1 - ˆp2) – 0/ ̂pˆq(1/n1 + 1/n2)
• RR:1) Reject H0 if z > z α
• 2) Reject H0 if z < -z α
• 3) Reject H0 if z > z α /2 or z < -z α /2
• Assumptions:
• Large sample(n1 p1≥ 5, n1q1 ≥5, n2 p2 ≥5, n2 q2 ≥5) • Samples are randomly and independently selected
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Example • Test the hypothesis that p1 - p2 < 0 if i t is known that
the test stat ist ic is
• z = -1.91.
• Solution:
• H0 : p1 - p2 = 0
• Ha : p1 - p2 < 0
• TS: z = -1.91
• RR: reject H0 if z < -1.645
• Dec: reject H0 • Conclusion: At 5% significance level there is sufficient
statistical evidence to conclude
• that p1 - p2 < 0.
Piyoosh Bajoria 21
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Reporting Results of Statistical Tests: P-Value• Definition. The p-value for a test of a hypothesis is the smallest
value of α for which the null hypothesis is rejected, i.e. the statistical
results are significant.• The p-value is called the observed significance level
• Note: The p-value is the probability ( when H0 is true) of obtaining avalue of the test statistic as extreme or more extreme than the actual
sample value in support of Ha.
• Examples. Find the p-value in each case:
• (i) Upper tailed test:H0 : θ = θ 0 ;Ha : θ> θ 0 ;
• TS: z = 1.76 p-value = .0392
• (ii) Lower tailed test:H0 : θ = θ 0 ;Ha : θ < θ 0
• TS: z = -1.86 p-value = .0314• (iii) Two tailed test: H0 : θ = θ 0 ;Ha : θ≠ θ 0
• TS: z = 1.76 p-value = 2(.0392) = .0784
• Decision rule using p-value: (Important)
• Reject H0 for all α > p- valuePiyoosh Bajoria 22