Introduction to Probability and Statistics
Sayantan Banerjee
Sessions 16 – 17
Testing of Hypothesis
• A vendor claims that his company fills any accepted order, on
the average, in at most 3 working days. You suspect that the
average is greater than that claimed. How will you go forward
with this?
• According to Fortune, on February 27, 2007, the average
stock in all U.S. exchanges fell by 3.3%. If a random sample
of 120 stocks reveals a drop of 2.8% on that day and a
standard deviation of 1.7%, are there grounds to reject the
magazine’s claim?
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Testing of Hypothesis
• What is a hypothesis?
• What do we mean by ‘testing’?
• Why do we need ‘testing’?
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The first step in hypothesis testing is to formalize it by specifying
the null and alternative hypothesis.
• Null hypothesis
I An assertion about the value of a population parameter.
I It is an assertion that we hold as true unless we have sufficient
statistical evidence to conclude otherwise.
I Generally denoted by H0 .
• Alternative hypothesis
I Negation of the null hypothesis.
I Researcher’s hypothesis.
I Generally denoted as H1 .
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Testing of Hypothesis
• A vendor claims that his company fills any accepted order, on
the average, in at most 3 working days. You suspect that the
average is greater than that claimed. How will you go forward
with this?
H0 : µ ≤ 3 vs. H1 : µ > 3.
• According to Fortune, on February 27, 2007, the average
stock in all U.S. exchanges fell by 3.3%. If a random sample
of 120 stocks reveals a drop of 2.8% on that day and a
standard deviation of 1.7%, are there grounds to reject the
magazine’s claim?
H0 : µ = 3.3 vs. H1 : µ 6= 3.3.
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• Although the idea of a null hypothesis is simple, determining
what the null hypothesis should be in a given condition is
difficult.
• Generally what the analyst aims to prove is the alternative
hypothesis, the null hypothesis standing for the status quo,
do-nothing situation.
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Examples
• A pharma company claims that 4 out of 5 doctors prescribe
their pain killer medicine. If you wish to test this claim, how
would you set up the hypotheses?
• It is found that web surfers will lose interest in a web page if
downloading takes more than 12 secs at 28K band rate. If you
wish to test the effectiveness of a newly designed web page in
regard to downloading time, how will you set up the
hypotheses?
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Testing of Hypothesis
• We DO NOT know the truth.
• Based on observed data, we either support H0 or H1 .
• Of course, we cannot be certain (so there is some room for
error).
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Type I and Type II errors
In the context of statistical testing of hypothesis, rejecting a true
null hypothesis is known as Type I error. Failing to reject a false
null hypothesis is known as a Type II error.
H0 true H0 false
Reject H0 Type I error X
Fail to reject H0 X Type II error
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Type I and Type II errors
Type-I error
• maximum P(Type-I error) is called ‘Type-I error rate’.
• maximum P(Type-I error) is denoted by α.
• α is also known as ‘level of significance’.
• P(Type-I error) is maximised at the boundary of H0 and H1 .
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Type I and Type II errors
Type-II error
• P(Type-II error) always computed at a particular θ in the
region of H1 .
• P(Type-II error) at a fixed θ denoted by β(θ).
• 1 − β(θ) also known as the ‘power’ of a test.
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Testing of Hypothesis
• Test Statistic: Statistic used to perform a specific testing of
hypothesis.
• Decision rule: Rule which decides which hypothesis to favour
on light of the value of the statistic.
• Critical region: Rejection region.
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Testing of Hypothesis
What is a ‘good’ test?
• Test which simultaneously minimizes both Type-I and Type-II
errors. But is this possible? (Why or why not?)
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Testing of Hypothesis
What is a ‘good’ test?
• Test which simultaneously minimizes both Type-I and Type-II
errors. But is this possible? (Why or why not?) NO!
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Testing of Hypothesis
What is a ‘good’ test?
• Test which simultaneously minimizes both Type-I and Type-II
errors. But is this possible? (Why or why not?) NO!
• We fix an upper bound for α, and then reduce the probability
of Type-II error.
• This guarantees that we cannot commit a larger Type-I error
than that specified.
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Testing population mean for Normal population
H0 : µ ≤ µ0 vs. H1 : µ > µ0 .
• Random sample X1 , . . . , Xn from N (µ, σ 2 ) population.
• Test statistic: X̄ ∼ N (µ, σ 2 /n).
• Observed test statistic: X̄obs
• Decision rule: Reject H0 if X̄obs > C.
• C is chosen so that maximum P(Type-I error) is
α = P (Reject H0 | µ = µ0 )
= P (X̄ > C | µ = µ0 ).
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Testing population mean for Normal population
How to find C?
• σ 2 known:
σ
C = µ0 + zα √ .
n
• σ 2 unknown:
S
C = µ0 + tα √ .
n
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What will be the decision rules for the following cases?
•
H0 : µ ≥ µ0 vs. H1 : µ < µ0 .
•
H0 : µ = µ0 vs. H1 : µ 6= µ0 .
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Testing population proportions
H0 : p ≤ p0 vs. H1 : p > p0 .
• Consider a random sample from Ber(p).
• Test statistic: X, the number of successes in n trials.
• Exact distribution: X ∼ Bin(n, p).
• Approx. distribution: X ∼ N (np, npq).
• Decision rule: Reject H0 if Xobs is too large.
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Testing population proportions
H0 : p ≤ p0 vs. H1 : p > p0 .
• Consider a random sample from Ber(p).
• Test statistic: X, the number of successes in n trials.
• Exact distribution: X ∼ Bin(n, p).
• Approx. distribution: X ∼ N (np, npq).
• Decision rule: Reject H0 if Xobs is too large.
• How large is large?
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Testing population proportions
H0 : p ≤ p0 vs. H1 : p > p0 .
• Decision rule: Reject H0 if (based on approx. distribution
under H0 )
Xobs − np0
√ ≥ C∗
np0 q0
• Given a significance level α,
C ∗ = zα .
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Testing population proportions
How to test the following:
•
H0 : p ≥ p0 vs. H1 : p < p0 .
•
H0 : p = p0 vs. H1 : p 6= p0 .
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Hypothesis testing: p-value approach
• p-value appoach
1. Cook up a reasonable test statistic (point estimator for θ)
2. Find its sampling distribution under H0 .
3. Find the p-value: Probability of getting extreme values under
H0 .
4. Reject H0 if p-value is small (p-value < α)
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p-value approach
H0 : p ≤ p0 vs. H1 : p > p0 .
• Consider a random sample from Ber(p).
• Test statistic: X, the number of successes in n trials.
• Exact distribution: X ∼ Bin(n, p).
• Approx. distribution: X ∼ N (np, npq).
• Compute the probability of extreme values
P (X ≥ Xobs | H0 )
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p-value approach
H0 : p ≤ p0 vs. H1 : p > p0 .
• Consider a random sample from Ber(p).
• Test statistic: X, the number of successes in n trials.
• Exact distribution: X ∼ Bin(n, p).
• Approx. distribution: X ∼ N (np, npq).
• Compute the probability of extreme values
P (X ≥ Xobs | H0 )
which is maximised at the boundary of H0 and H1
p − value = P (X ≥ Xobs | p = p0 )
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p-value approach (contd.)
H0 : p ≤ p0 vs. H1 : p > p0 .
• Decision rule: Reject H0 if p − value ≤ α.
• Reject H0 if X ≥ Xobs is less likely under H0
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p-value approach
Testing Normal means
H0 : µ ≤ µ0 vs. H1 : µ > µ0 .
H0 : µ ≥ µ0 vs. H1 : µ < µ0 .
H0 : µ = µ0 vs. H1 : µ 6= µ0 .
Find p-value for the tests based on the test statistic X̄, both for
the cases where σ is known and unknown.
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Problems
According to BusinessWeek, the average market value of a biotech
company is less than $250 million. A random sample of 30 firms
reveal a mean of $235 million with sd $85 million. Use α = 0.05 to
test this claim, and state your conclusions.
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Problems
According to Fortune, on February 27, 2007, the average stock in
all U.S. exchanges fell by 3.3%. If a random sample of 120 stocks
reveals a drop of 2.8% on that day and a standard deviation of
1.7%, are there grounds to reject the magazine’s claim?
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Problems
In a random sample of 30 students at IIMI it was found that 22 use
non-Apple laptops. Test at α = 0.05 if there is sufficient evidence
to conclude that over 60% of the students use non-Apple laptops.
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Problems
A random sample of 150 recent donations at a certain blood bank
reveals that 82 were type A blood. Does this suggest that the
actual percentage of type A donations differs from 40%, the
percentage of the population having type A blood? Carry out a
test of the appropriate hypotheses using a significance level of
0.01. Would your conclusion have been different if a significance
level of 0.05 had been used?
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