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Hypothesis Tests
In statistics a hypothesis is a statement that
something is true.
Hypothesis Tests
In statistics a hypothesis is a statement that
something is true.
For example:
Using a only person’s horoscope, a
professional astrologer can correctly predict
which of three personality charts applies to
that person with probability 1/3.
Hypothesis Tests
For example:
Using a only person’s horoscope, a
professional astrologer can correctly predict
which of three personality charts applies to
that person with probability 1/3.
Another example:
A (unbiased) jury assume a person is not
guilty until proven innocent.
Hypothesis Tests
The Null Hypothesis, H0, is a statement
about values of a population parameter of a
population. This is normally the status quo
and it contains an equality.
The Alternate (Research) Hypothesis, HA,
is a statement that is true when the Null
Hypothesis is false.
Hypothesis Tests
If you make a claim that is different from the
status quo, then that is the alternative
hypothesis.
If a 3rd party makes a claim:
– If the claim contains equality, then it is the null
hypothesis
– If the claim does not contain equality it is the
alternative hypothesis.
Hypothesis Tests
In a hypothesis test:
1. Identify H0 and HA
2. Select a level of significance ( )
3. Assume the null hypothesis is true
4. Take a sample and determine the
probability of that occurring. This is
called the p-value.
5. Reject or Fail to reject H0
Example
A local brewery distributes beer in bottles
labeled 341 ml. A group of college students
randomly selected a sample of 50 bottles and
obtained a mean of 339 ml and a standard
deviation of 4.8 ml. The college students claim
that the brewery is under-filling their bottles.
How probable is a sample mean of 339 or
lower? (Use   .05)
H0: m341
over-filling
HA: m<341
under-filling
Example
n  50,
x  339,
s  4.8


 x  m 339  341 
P x  339   P
<


4
.
8


n
50 

 P z < 2.95
 0.5  P0  z  2.95
 .0016
It is highly unlikely to obtain a sample mean of 339, the fact
that we obtained it gives sufficient evidence to reject the null
hypotheses, that is, there is sufficient evidence to support the
claim that the brewery is underfilling their bottles.
Example
A poll of 100 randomly selected car owners
revealed that the mean length of time that
they plan to keep their cars is 7.01 years and
the standard deviation is 3.74 years. The
president of the Trebor Car Park is trying to
plan a sales campaign targeted at car owners
who are ready to buy a different car. Test the
claim of the sales manager, who
authoritatively states that the mean time all
car owners plan to keep their cars is less than
7.5 years. Use a 0.05 significance level.
Hypothesis Tests
That was a one tailed test. Examples of one
tailed tests:
H0: m  2400 HA: m  2400 OR
H0: m  7.51
HA: m < 7.51
In a two tailed test the alternate hypothesis
contains inequality. For example:
H0: m  550
HA: m  550
Hypothesis Tests
In a two tailed hypothesis test:
1. Identify H0 and HA
2. Select a level of significance ( )
3. Assume the null hypothesis is true
4. Take a sample and determine the
probability of that occurring and multiply
it by two. This is the p-value.
5. Reject or Fail to reject H0
Example
Suppose that we want to test the hypothesis with a
significance level of .05 that the climate has
changed since industrialization. Suppose that the
mean temperature throughout history is 50
degrees. During the last n=40 years, the mean
temperature has been 51 degrees with a standard
deviation of 2 degrees. What can we conclude?
H0: m  50
HA: m  50
Rejection Regions
Suppose that  = .05. We can draw the
appropriate picture and find the z score for:
 / 2. We call the outside regions the
rejection regions.
Rejection Regions
We call the blue areas the rejection region
since if the value of z falls in these regions,
we reject the null hypothesis
Example
50 smokers were questioned about the number of
hours they sleep each day. Test the hypothesis
that the smokers need less sleep than the general
public which needs an average of 7.7 hours of
sleep. Compute a rejection region for a
significance level of .05. If the sample mean is 7.5
and the standard deviation is .5, what can you
conclude?
H0:
m  7. 7
HA:
m < 7. 7
Hypothesis Tests (with rejection
regions)
1.
2.
3.
4.
5.
Identify H0 and HA
Select a level of significance ( )
Assume the null hypothesis is true
Find the rejection region
Take a sample and determine the
corresponding z-score
6. Reject or Fail to reject H0
Exercises
• #8.8, 8.11, 8.13 on page 375
• #8.42, 8.44, 8.49, 8.52 page 389
• #8.26, 8.34 on page 381
Homework
• Review Chapters 8.1-8.3
• Read Chapters 8.4, 8.5, 8.7
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