Download Null Hypothesis H 0

Business Statistics for Managerial
Decision
Farideh Dehkordi-Vakil
Tests of Significance
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Confidence intervals are appropriate when our goal is to
estimate a population parameter.
The second type of inference is directed at assessing the
evidence provided by the data in favor of some claim about
the population.
A significance test is a formal procedure for comparing
observed data with a hypothesis whose truth we want to
assess.
The hypothesis is a statement about the parameters in a
population or model.
The results of a test are expressed in terms of a probability
that measures how well the data and the hypothesis agree.
Example: Bank’s net income
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The community bank survey described in previous
lecture also asked about net income and reported
the percent change in net income between the first
half of last year and the first half of this year. The
mean change for the 110 banks in the sample
is X  8.1% Because the sample size is large, we
are willing to use the sample standard deviation
s = 26.4% as if it were the population standard
deviation . The large sample size also makes it
reasonable to assume that X is approximately
normal.
Example: Bank’s net income
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Is the 8.1% mean increase in a sample good evidence that
the net income for all banks has changed?
The sample result might happen just by chance even if the
true mean change for all banks is  = 0%.
To answer this question we asks another
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Suppose that the truth about the population is that = 0% (this is
our hypothesis)
What is the probability of observing a sample mean at least as far
from zero as 8.1%?
Example: Bank’s net income
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The answer is:
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p( X  8.1)  P( Z 
8.1  0
)  P( Z  3.22)
26.4 110
 1  .9994  .0006
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Because this probability is so small, we see that the
sample mean X  8.1 is incompatible with a population
mean of  = 0.
We conclude that the income of community banks has
changed since last year.
Example: Bank’s net income
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The fact that the calculated probability is very
small leads us to conclude that the average percent
change in income is not in fact zero. Here is why.
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If the true mean is  = 0, we would see a sample mean
as far away as 8.1% only six times per 10000 samples.
So there are only two possibilities:
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 = 0 and we have observed something very unusual, or
 is not zero but has some other value that makes the
observed data more probable
Example: Bank’s net income
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We calculated a probability taking the first
of these choices as true ( = 0 ). That
probability guides our final choice.
If the probability is very small, the data
don’t fit the first possibility and we
conclude that the mean is not in fact zero.
Example:Is this percent change
different from zero?
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Suppose that next year the percent change in net
income for a sample of 110 banks is X = 3.5%.
(We assume that the standard deviation  =
26.4%.) This sample mean is closer to the value
 = 0 corresponding to no mean change in income.
What is the probability that the mean of a sample
of size n = 110 from a normal population with
 = 0 and standard deviation  = 26.4 is as far
away or farther away from zero as X  3.5 ?
Example:Is this percent change
different from zero?
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The answer is:
P( X  3.5)  P( z 
3.5  0
)  P( z  1.39)
26.4 110
 1  .9177  .08
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A sample result this far from zero would happen just by
chance in 8% of samples from a population having true
mean zero.
An outcome that could so easily happen by chance is
not good evidence that the population mean is different
from from zero.
Example:Is this percent change
different from zero?
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The mean change in net assets
for a sample of 110 banks will
have this sampling distribution
if the mean for the population
of all banks is  = 0.
A sample mean X  3.5%could
easily happen by chance. A
sample mean X  8.1% is far out
on the curve that it would rarely
happen just by chance.
Tests of Significance: Formal details
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The first step in a test of significance is to state a
claim that we will try to find evidence against.
Null Hypothesis H0
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The statement being tested in a test of significance is
called the null hypothesis.
The test of significance is designed to assess the
strength of the evidence against the null hypothesis.
Usually the null hypothesis is a statement of “no effect”
or “no difference.” We abbreviate “null hypothesis” as
H0.
Tests of Significance: Formal details
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A null hypothesis is a statement about a population,
expressed in terms of some parameter or parameters.
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The null hypothesis in our bank survey example is
H0 :  = 0
It is convenient also to give a name to the statement we
hope or suspect is true instead of H0.
This is called the alternative hypothesis and is abbreviated
as Ha.
In our bank survey example the alternative hypothesis
states that the percent change in net income is not zero. We
write this as
Ha :   0
Tests of Significance: Formal details
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Since Ha expresses the effect that we hope to find evidence
for we often begin with Ha and then set up H0 as the
statement that the Hoped-for effect is not present.
Stating Ha is not always straight forward.
It is not always clear whether Ha should be one-sided or
two-sided.
 The alternative Ha :   0 in the bank net income
example is two-sided.
 In any given year, income may increase or decrease, so
we include both possibilities in the alternative
hypothesis.
Example:Have we reduced processing
time?
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Your company hopes to reduce the mean time 
required to process customer orders. At present,
this mean is 3.8 days. You study the process and
eliminate some unnecessary steps. Did you
succeed in decreasing the average process time?
You hope to show that the mean is now less than
3.8 days, so the alternative hypothesis is one
sided, Ha :  < 3.8. The null hypothesis is as usual
the “no change” value, H0 :  = 3.8.
Tests of Significance: Formal details
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Test statistics
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We will learn the form of significance tests in a
number of common situations. Here are some
principles that apply to most tests and that help
in understanding the form of tests:
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The test is based on a statistic that estimate the
parameter appearing in the hypotheses.
Values of the estimate far from the parameter value
specified by H0 gives evidence against H0.
Tests of Significance: Formal details
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A test statistic measures compatibility
between the null hypothesis and the data.
Many test statistics can be thought of as a
distance between a sample estimate of a
parameter and the value of the parameter
specified by the null hypothesis.
Example: bank’s income
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The hypotheses:
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H0 :  = 0
Ha :   0
The estimate of  is the sample mean X .
Because Ha is two-sided, large positive and
negative values of X (large increases and
decreases of net income in the sample) counts
as evidence against the null hypothesis.
Example: bank’s income
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The test statistic
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The null hypothesis is H0 :  = 0, and a sample gave
the X  8.1 . The test statistic for this problem is the
standardized version of X :
z
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X  0
 n
This statistic is the distance between the sample mean
and the hypothesized population mean in the standard
scale of z-scores.
z
8.1  0
 3.22
26.4 110
Tests of Significance: Formal details
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The test of significance assesses the evidence against the
null hypothesis and provides a numerical summary of this
evidence in terms of probability.
P-value
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The probability, computed assuming that H0 is true, that the test
statistic would take a value extreme or more extreme than that
actually observed is called the P-value of the test. The smaller the
p-value, the stronger the evidence against H0 provided by the data.
To calculate the P-value, we must use the sampling distribution of
the test statistic.
Example: bank’s income
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The P-value
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In our banking example we found that the test statistic
for testing H0 :  = 0 versus Ha :   0 is
z
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8.1  0
 3.22
26.4 110
If the null hypothesis is true, we expect z to take a value
not far from 0.
Because the alternative is two-sided, values of z far
from 0 in either direction count ass evidence against H0.
So the P-value is:
P( z  3.22)  p ( z  3.22)
 (1  .9994)  0.0006  .0012
Example: bank’s income
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The p-value for bank’s
income.
The two-sided p-value is
the probability (when H0
is true) that X takes a
value at least as far from 0
as the actually observed
value.
Tests of Significance: Formal details
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We know that smaller P-values indicate stronger
evidence against the null hypothesis.
But how strong is strong evidence?
One approach is to announce in advance how
much evidence against H0 we will require to reject
H0.
We compare the P-value with a level that says
“this evidence is strong enough.”
The decisive level is called the significance level.
It is denoted be the Greek letter .
Tests of Significance: Formal details
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If we choose  = 0.05, we are requiring that
the data give evidence against H0 so strong
that it would happen no more than 5% of
the time (1 in 20) when H0 is true.
Statistical significance
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If the p-value is as small or smaller than , we
say that the data are statistically significant at
level .
Tests of Significance: Formal details
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You need not actually find
the p-value to asses
significance at a fixed
level .
You need only to compare
the observed statistic z
with a critical value that
marks off area  in one or
both tails of the standard
Normal curve.
Test for a Population Mean