Transcript Introductory Mathemathematics & Statistics for Business

Chapter S12
Hypothesis testing
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Learning Objectives
– Understand the principles of statistical inference
– Formulate null and alternative hypotheses
– Understand one-tailed and two-tailed tests
– Understand type I and type II errors
– Understand test statistics
– Understand the significance level of a test
– Understand and calculate critical values
– Understand the regions of acceptance and rejection
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Chapter S12
Hypothesis testing
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Learning Objectives continued ...
– Calculate and interpret a one-sample z-test statistic
– Calculate and interpret a one-sample t-test statistic
– Calculate and interpret a paired t-test statistic
– Calculate and interpret a two-sample t-test statistic
– Understand and calculate p-values
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Statistical inference
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One of the major roles of statisticians is to draw
conclusions from data
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This is referred to as statistical inference
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We can put a probability on whether a conclusion is
correct within reasonable doubt
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Conclusions can always be wrong
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Statistical inference plays a major role in decision
making
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Statistical inference
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Decision-making process:
1.Collect the data
2. Summarise the data (using either visual displays or
descriptive statistics)
3.Set up an hypothesis (i.e. claim or theory) to be tested
4.Calculate the probability of obtaining a sample such as
the one we have if the hypothesis is true
5.Either accept or reject the hypothesis
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Conclusions will be made based on samples taken
from the population
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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The null hypothesis
(questions dealing with differences between
samples)
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Technique for dealing with these problems begins
with the formulation of an hypothesis.
Null hypothesis is a statement that nothing
unusual has occurred. The notation is Ho.
Alternative hypothesis states that something
unusual has occurred. The notation is H1 or HA
Together they may be written in the form:
Ho:(statement) v H1(alternative statement)
Where: ‘v’ stands for versus
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Alternative hypothesis
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May be classified as two-tailed test or one-tailed test
Two-tailed test (two sided alternative)
Test with no preconceived notion that the true value
of µ is either above or below the hypothesised value
of µ
H1: µ  µo
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Errors
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There are two possible errors in making
a conclusion about a null hypothesis
1. Type I errors occur when you reject Ho as
being false when Ho is really true
2. Type II errors occur when you accept Ho as
being true when Ho is really false
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Significance level
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This level represents the borderline probability
between whether an event has occurred by chance
or whether an unusual event has taken place
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Most common significance level used is 0.05,
commonly written as  = 0.05
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5% significance level says in effect that an event that
occurs less than 5% of the time is considered unusual
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Test statistics
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Test for determining if a single sample from a
population is consistent with the rest of the population
• one sample z-test
• one sample t-test
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Value of a test statistic
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Information required in calculation:
1. the size (n) of the sample
2. the mean ( x ) of the sample
3. the standard deviation (s) of the sample
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Other information of interest might include:
1. Does the population have a normal distribution?
2. Is the population’s standard deviation known?
3. Is the sample size (n) large?
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Drawing a conclusion
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The steps that should be undertaken to perform a
one-sample test are:
1. Set up null and alternative hypotheses—including deciding
whether to use a one-sided or two-sided test
2. Decide on significance level you are using
3. Write down the relevant data
4. Decide on the test statistic to be used
5. Calculate the value of the test statistic
6. Find the relevant critical value and decide whether H0 is to
be not rejected or rejected
7. Draw an appropriate conclusion
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Test statistics
(two sample problems)
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Paired t-test
– two samples that are to be compared with each
other
– often referred to as two-sample problems
– have a structure such that the data are paired
– samples must each contain the same number of
observations
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Test statistics
(two sample problems)
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Two-sample t-test
– Samples are not paired, they are independent
– Two samples need not contain the same number
of observations
– Most common test statistic used in this situation is
a two-sample t-test
– Also known as a pooled test
– Calculation requires more work than for the paired
t-test
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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Test statistics
(two sample problems)
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p-values
– Alternative to using critical values for testing
hypotheses
– Calculates the probability of obtaining a value as
extreme as the value of the test statistic
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
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