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STAT 231
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Matthias Schonlau
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Statistics

STAT 231

Matthias Schonlau

Fall

Description

Chapter 4
Testing Statistical Hypotheses
Matthias Schonlau
Stat 231 Overview
• Introduction
• Testing statistical hypotheses
– Example T-test
• Significance levels
– Example: Spending in Coffee shops
• One sided alternative Hypotheses
– Example: Survivor 2011
– Example: Nadal vs Murray Introduction
• The purpose of statistical inference is to draw
conclusions about a population parameter
– Point estimates
• What is the “best” estimate ? (maximum likelihood)
– Confidence intervals
• What are plausible values for the population
parameter?
– Hypothesis tests
• Test whether the population parameter could be a
specific value Introduction
• In law we are used to making hypotheses:
H : Defendant is innocent
0
H1: Defendant is guilty
• H zero is pronounced “H naught”
• We find the defendant “guilty” or “not guilty”
• We cannot prove the defendant is innocent. Introduction
• Legal trials use the concept of “beyond
reasonable doubt”
– If a suspect is acquitted, this does not mean he/ she
didn’t do it.
– It means the jury was not certain “beyond a
reasonable doubt”
• In the US, for the same crime you can be found
– not guilty in a criminal trial but
– guilty in the civil trial
The criminal trial requires a greater degree of evidence Testing statistical Hypotheses
• Statistical hypotheses test whether a
population parameter could be a specific
value
µ 0
H 0 0 µµ=
H 1 0 µµ≠ Testing statistical hypotheses
• In deductive logic we establish have whether
something is true or not.
• In statistics, we never know for sure whether a
hypothesis is correct or not.
• We can establish whether the hypothesis is
very unlikely to occur by chance giventhe
data
• Given the hypothesis, what is the probability
of observing dobsr something more extreme?
p =D(d ≥ obs o Testing statistical hypotheses
• The famous t-test
• Assume YG~ (,)µσ
• Calculate the probability or p-value
PT tT≥t~ ) n−
• where
y −µ
t = 0
s/
• t measures the distance of the hypothesized
value to
y Example T-test
• A toy example with only 3 observations.
• Test whether the population mean is 0.
H 0 µ =0
H : µ ≠0
1 Example T-test
• Compute the t-statistic:
• n-1=2 degrees of freedom
• What is the probability of getting a |t| value of
4.157 or larger? Example T-test
t-dist->red (df = 2) normal dist->blue
-5 -4 -3 -2 -1 0 1 2 3 4 5
Critical value of t = ±4.16 (alpha=.0533, 2-tail)
• What is the probability of getting a |t| value of
4.157 or larger? Example T-test
• To get the probability, we need a computer:
• The probability is 5.3%. Example T-test
• From the t-table in the notes, we know
t2,α=0.054.3
t = 2.92
2,α=0.10
• The observed value is in the bounded by these
two values.
• Therefore we know the probability is between
5% and 10%. Example T-test
• We reject the null-hypothesis, if the observed
statistics is unlikely given the null hypothesis.
• Small p-values are strong evidence against the
hypothesis
• Is 5.3% unlikely enough? Testing statistical hypotheses
• There are thresholds that are typically used
– those threshold are called significance levels Significance levels
Evidence levels against the hypothesis
Significance levels Interpretation
p>0.1 No evidence
0.05 One –sided alternative hypotheses
• If a new drug is introduced we would like to
know whether the new drug performs better
than the current one.
H 0 effectiveness new drug ≤ effectiveness old drug
H : effectivness new drug > effectieness old drug
1
• This is called a one sided hypotheses One –sided alternative hypotheses
H 0 0 µµ= Shaded area is the probability of
H : µµ≠ a more extreme sample average)or
1 0 under the null hypothesis One –sided alternative hypotheses
H 0 0 µµ≤ Shaded area is the probability of
H : µµ> a more extreme sample average)or
1 0 under the null hypothesis One –sided alternative hypotheses
H 0 0 µµ≥ observing the sample average (orf
H : µµ< a more extreme sample average)
1 0 under the null hypothesis One –sided alternative hypotheses
Thes

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