Download Section 6.2

Section 6.2
Introduction to
Hypothesis Testing
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•An important technique in the inferential statistics is
hypothesis testing.
A hypothesis is a process that uses the sample statistic
to test the population parameter.
Researchers in medicine, psychology and business rely
on hypothesis testing to make informed decisions about
new medicines, treatments and marketing strategies.
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Suppose a cellular phone company advertises that the mean talk of
a new cellular phone battery is 180 minutes. If you suspect that
the mean talk time is not 180 minutes, how could you show that
the ad is false?
We could take a random sample from the population of batteries
and measure the talk time of each. If the sample mean differs
enough then we can decide that the ad is wrong or false.
µ = 180
For instance, to test the mean talk time of all cellular phone
batteries
You could take a random sample of n = 100 batteries and measure
the talk time of each. Suppose you obtain a sample mean of 174
minutes with a standard deviation of s =20 min, does this
indicate that the company’s ad is false.
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n
n
n
n
To decide, you do something unusual. We
assume that the ad µ = 180 is correct!
Then you examine the sampling distribution of
the sample means (with n= 100) taken from the
population in which mean µ = 180 and σ =20.
From the Central limit theorem we know that
this sampling distribution is normal with mean
of 180 and standard deviation of
20/sqrt(100) = 2
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0.25
0.2
0.15
0.1
0.05
0
168
170
172
174
176
178
180
182
184
186
188
190
192
-0.05
n
The probability of obtaining 172 or less is a rare
occurrence! You assumption that the company’s ad is
right has led to an improbable result. So either you had
an unusual sample or the ad is probably false. The
logical conclusion is that the ad is false.
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A statistical hypothesis is a claim about a
population.
Null hypothesis H0
contains a statement of
equality such as ≥ , = or ≤.
Alternative hypothesis Ha
contains a statement of
inequality such as < , ≠ or >
Complementary Statements
If I am false,
you are true
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If I am false,
you are true
Writing Hypotheses
Write the claim about the population. Then, write its complement.
Either hypothesis, the null or the alternative, can represent the
claim.
A hospital claims its ambulance response time is less
than 10 minutes.
claim
A consumer magazine claims the proportion of cell
phone calls made during evenings and weekends is at
most 60%.
claim
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Hypothesis Test Strategy
Begin by assuming the equality condition in the null hypothesis is true.
This is regardless of whether the claim is represented by the null
hypothesis or by the alternative hypothesis.
Collect data from a random sample taken from the population
and calculate the necessary sample statistics.
If the sample statistic has a low probability of being drawn
from a population in which the null hypothesis is true, you will
reject H0. (As a consequence, you will support the alternative
hypothesis.)
If the probability is not low enough, fail to reject H0.
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Error possible
n
n
n
n
When you perform the hypothesis test, you
make one of the two decisions.
A. Reject null hypothesis
B. Fail to reject the null hypothesis
Remember that your decision is based on a
sample rather than the entire population, so
there is always a possibility that you make the
wrong decision.
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Decision
Errors and Level of Significance
Actual Truth of H0
Do not
reject H0
Reject H0
H0 True
Correct
Decision
Type I
Error
H0 False
Type II
Error
Correct
Decision
A type I error: Null hypothesis is actually
true but the decision is to reject it.
Level of significance,
Maximum probability of committing a type I error.
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American Justice System
verdict
Truth about defendant
Innocent
Not guilty
Reject H0
H0 False
Justice
Type II
Error
Type I
Error
Justice
A type I error: Null hypothesis is actually
true but the decision is to reject it.
Level of significance,
Maximum probability of committing a type I error.
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