The Nature of Hypothesis Testing

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The Nature of Hypothesis Testing
Null Hypothesis Hypothesis to be tested. We
represent the null hypothesis by using
Alternative Hypothesis A hypothesis to be
considered as an alternative to the null
hypothesis.
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The Nature of Hypothesis Testing
A manufacturer claims that the mean amount of
juice in its 16 ounce bottles is 16.1 ounces. A
consumer advocacy group wants to perform a
hypothesis test to determine whether the mean
amount is actually less then this.
Determine the null and Alternative Hypotheses.
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The Nature of Hypothesis Testing
Alternative Hypotheses Options
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The Nature of Hypothesis Testing
A health insurer has determined that the
reasonable and customary fee for a certain
medical procedure is 1200. They suspect that the
average fee charged by one particular clinic for
this procedure is higher then 1200. The insurer
wants to perform a hypothesis test to determine
whether their suspicion is correct.
Determine the null and alternative hypothesis and
indicate what type of alternative hypothesis.
Right-tailed hypothesis
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The Nature of Hypothesis Testing
Basic logic of hypothesis testing
Take a random sample from the population. If the
sample is consistent with the null hypothesis, do
not reject the null hypothesis.
If the sample data are inconsistent with the null
hypothesis, reject the null hypothesis and
conclude that the alternative hypothesis is true.
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Terms, Errors, and Hypotheses
Test Statistic The statistic used as a basis for
deciding whether the null hypothesis should be
rejected.
Rejection Region The set of values for the test
statistic that leads to rejection of the null
hypothesis.
Nonrejection Region The set of values for the
test statistic that leads to nonrejection of the
null hypothesis.
Critical Values The values of the test statistic
that separate the rejection and nonrejection
regions. A critical value is part of the
rejection region.
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Terms, Errors, and Hypotheses
Type I error Rejecting the null hypothesis when
it is true.
Type II error Not rejecting the null hypothesis
when it is false.
The mean running time for a certain type
flashlight battery is 8.5 hours. The manufacturer
has introduced a change in the production method
and wants to perform a hypothesis test to
determine whether the mean running time has
increased as a result. The hypotheses are
Explain the meaning of a type I and a type II
error.
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Terms, Errors, and Hypotheses
Significance Level The probability of making a
type I error, that is, of rejecting a true null
hypothesis , is called the significance level,
alpha, of a hypothesis test.
Relation Between type I and Type II Error
Probabilities
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Terms, Errors, and Hypotheses
Possible Conclusions for a Hypothesis Test
Suppose a hypothesis test is conducted at a small
significance level
1. If the null hypothesis is rejected, we
conclude that the alternative hypothesis is true.
2. If the null hypothesis is not rejected, we
conclude that the data do not provide sufficient
evidence to support the alternative hypothesis.
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Hypothesis Tests for 1 Population Mean
Obtaining Critical Values
Suppose that a hypothesis test is to be performed
at the significance level, alpha. Then any
critical values must be chosen so that, if the
null hypothesis is true, the probability is alpha
that the test statistic will fall in the
rejection region.
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Hypothesis Tests for 1 Population Mean
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Hypothesis Tests for 1 Population Mean
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Hypothesis Tests for 1 Population Mean
The mean length of imprisonment for motor-vehicle
theft offenders in Australia is 16.7 months. 100
randomly selected motor-vehicle theft offenders
in Sydney had a mean length of imprisonment of
17.8 months. At the 5 significance level, do the
data provide sufficient evidence to conclude that
the mean length of imprisonment for these
offenders in Sydney differs from the national
mean in Australia. Assume the the population
standard deviation of the lengths of imprisonment
for motor-vehicle theft offenders in Sydney is
6.0 months.
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Hypothesis Tests for 1 Population Mean
Step 1 Determine null and alternative hypothesis.
Step 2 Decide on the significance level alpha.
Step 3 Compute the test statistic Z.
Step 4 Determine the type of test.
Step 5 If the test statistic falls in the
rejection region we reject the null hypothesis
otherwise we do not reject.
Step 6 Interpret the results of the hypothesis
test.
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Hypothesis Tests for 1 Population Mean
Step 6 At the 5 significance level, the data do
not provide sufficient evidence to conclude that
the mean length of imprisonment, mu, of
motor-vehicle theft offenders in Sydney differs
from the national mean in Australia.
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Hypothesis Tests for 1 Population Mean
Presume we have last years local monthly bills
for a random sample of 50 cell phone users. At
the 1 significance level, do the data provide
sufficient evidence to conclude that last years
mean bill has decreased from the 2001 mean of
47.37. Assume the standard deviation is 25. The
sum of the 50 phone bills is 2053.48.
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Hypothesis Tests for 1 Population Mean
Step 6 At the 1 significance level, the data do
not provide sufficient evidence to conclude that
the mean cell phone bill has decreased since
2001.
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Hypothesis Tests for 1 Population Mean
Researchers obtained the body temperatures of 93
healthy humans. At the 1 significance level, do
the data provide sufficient evidence to conclude
that that the mean body temperature of healthy
humans differs from 98.6 F? Assume that the
standard deviation is 0.63 degrees F and the sum
of the temperatures is 9125.5.
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Hypothesis Tests for 1 Population Mean
Step 6 At the 1 significance level, the data
provide sufficient evidence to conclude that the
mean body temperature of all healthy humans from
the accepted value of 98.6.
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P-Values
P-value indicates how likely observation of the
value obtained in the test statistic would be if
the null hypothesis is true.
The smaller the p-value, that is, closer to 0,
the stronger the evidence against the null
hypothesis.
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P-Values
The bookstore sold history books for a mean of
51.46 in 2000. Were to determine if the mean
has increased.
Obtain and interpret the P-value of the
hypothesis test.
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P-Values
The hypothesis is a right-tailed z-test.
Therefore the P-value is the probability of
observing a z of 2.85 or better if the null
hypothesis is true.
That probability equals the area under the
standard normal curve to the right 2.85. Diagram
to the right below.
From table II we find the area to be 1 0.9978
0.0022
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P-Values
Therefore, the P-value of this hypothesis test is
0.0022. If the null hypothesis is true we would
observe a value of the test statistic z of 2.85
or greater only about 2 times in a thousand.
If the null hypothesis is true a sample of 40
history books would have a mean of 54.89 or
greater about 0.2 of the time.
The data provide strong evidence against the null
hypothesis.
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P-Values
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P-Values
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P-Values
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Hypotheses Tests when Standard Deviation is
Unknown
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Hypotheses Tests when Standard Deviation is
Unknown
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Hypotheses Tests when Standard Deviation is
Unknown
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P-Values
The mean annual consumption of beer per person in
the U.S. is 22.0 gallons (about 235 12 oz.
bottles. A random sample of 300 D.C. residents
yielded a mean annual beer consumption of 27.8
gallons. At the 10 significance level, do the
data provide sufficient to conclude that the mean
annual consumption of beer per person for the
nations capital differs from the national mean?
Assume the standard deviation for D.C. residence
is 55 gallons.
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P-Values
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P-Values
Step 6 At the 10 significance level, the data
provide sufficient evidence to conclude that the
mean annual beer consumption by D.C. residents
differs from the national mean of 22.0 gallons.