Stat 104 – Lecture 23
Testing Hypotheses
• Hypothesis – a tentative theory.
• Statistical hypothesis – a tentative
theory about the value of a population
parameter.
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Testing Hypotheses
• Use sample data, in the form of a
sample statistic, to support or
refute hypotheses.
• How likely is it to get the sample
data if a hypothesis is true?
2
Example
• The hog processing plant wants to
see if the population mean hot
carcass weight of hogs is 200 lbs or
is it something less than 200 lbs.
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Stat 104 – Lecture 23
Example
• Null hypothesis
– H0:μ = 200
• Alternative hypothesis
– HA:μ < 200
•
μ is the population mean hot
carcass weight.
4
Decisions & Conclusions
• Reject H0 – the population mean is
a value given in the alternative
hypothesis.
• Fail to Reject H0 – the population
mean could be the value given in
the null hypothesis.
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Errors
H0 is True
Fail to reject
H0
Reject H0
H0 is False
Correct
Decision
Type II Error
Type I Error
Correct
Decision
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Stat 104 – Lecture 23
Errors
• Would like to reduce the chance of
making either error.
• One way to do this is to have a
large sample of data.
7
Level of Significance
• Denoted by the Greek letter, α
• The probability of making a Type I
error.
• Typically,α is chosen to be 5% or
0.05.
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Test Statistic
• Summary of the sample data used
to make a decision in a test of
hypothesis.
• Depends on what is known about
the population.
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Stat 104 – Lecture 23
Testing – Step by Step
•
•
•
•
•
Step 1 – Set-Up
Step 2 – Test Criteria
Step 3 – Sample Evidence
Step 4 – Probability Value
Step 5 – Results
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Step 1 – Set-Up
•
μ is the population mean hot
carcass weight of hogs.
• Null hypothesis
– H0:μ = 200
• Alternative hypothesis
– HA: μ < 200
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Step 2 – Test Criteria
• Distribution of the sample mean is
approximately normal.
• σ = 26.1 lbs
• z test statistic.
• α = 0.05
12
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Stat 104 – Lecture 23
Step 3 – Sample Evidence
• Sample mean, y = 187.5
• Test statistic,
z=
y − μ0
⎛
⎜
⎝
σ
n
⎞
⎟
⎠
=
187.5 − 200 − 12.5
=
= −3.39
3.961
⎛ 26.1 ⎞
⎜
⎟
⎝ 50 ⎠
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Step 4 – Probability Value
• Alternative hypothesis
– HA:μ < 200
• P-value is the probability of getting
a value of z smaller than –3.39
• Table Z: 0.0003
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Step 5 - Results
• Reject the null hypothesis because
the P-value is smaller than 0.05
• The population mean hot carcass
weight of hogs is less than 200 lbs.
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