TESTING OF HYPOTHESIS
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Outline
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Testing a Claim About a Mean µ , When Known
Testing a Claim About a Mean µ, When Not Known
Testing a Claim About a Proportion
Testing a Claim About a Standard Deviation or Variance
Types of Errors
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A Type I Error occurs if we reject the null hypothesis when it is
true.
A Type II error occurs if we fail to reject the null hypothesis if it is
false.
Example
A type I error is analogous to convicting an innocent person for a
crime they didn’t commit.
A type II error is analogous to failing to convict a guilty person.
Type I & Type II Error
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Referring
to Ho, the
Null
Hypothesis
True False
Reject Type I O.K
Error
Fail to O.K. Type II
Reject Error
Level of Significance
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The level of significance is the probability of rejecting the null hypothesis
when it is true.
A common level of significance is .05 (that means if we reject the null hypothesis,
we will be at least 95% sure that the null hypothesis is false).
We will reject the null hypothesis if P-value ≤
If P-value > , we do not reject the null hypothesis
Summary of Hypothesis Tests
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Determine the null and alternative hypothesis and set the level of
significance
Collect the data and compute the test statistic
Compute the P-value
If P-value ≤ , then reject H0; If P-value > , then do not reject H0
Example
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Critical Region (or Rejection Region)
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Set of all values of the test statistic that would cause a rejection of the
null hypothesis
Right-tailed Test
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H0: =
H1: > Points Right
Values that
differ significantly
from Ho
Left-tailed Test
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H0: =
H1: <
Points Left
Values that
differ significantly
from Ho
Critical Region Method
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As with previous method for hypothesis tests, determine H0, H1 and .
Instead of waiting to compute P-value and compare to , you
predetermine the critical region, that is the values of the test statistic
at which you will reject H0.
Then compute test statistic, and if it is in the critical region, reject H0
otherwise do not reject H0 .
Example
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The mean and standard deviation 17.09 and 3.87 (respectively).
H1: µ ≠ 17.09
H0: µ = 17.09
At a 5% significance level (i.e. α = .05), we have
α /2 = .025 Thus, z.025 = 1.96 and our rejection region is:
z < –1.96 -or- z > 1.96
z
-z.025 0 +z.025
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The Estimated mean = 17.55 from a sample of 100 observation
Using our standardized test statistic:
We find that:
Since z = 1.19 is not greater than 1.96, nor less than –1.96 we cannot
reject the null hypothesis in favor of H1.
Example
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ˆ
A survey of n = 880 randomly selected adult drivers showed
that 56% of those respondents admitted to running red lights.
Find the value of the test statistic for the claim that the
majority of all adult drivers admit to running red lights.
Solution16
The preceding example showed that the given claim results in
the following null and alternative hypotheses:
H0: p = 0.5 and H1: p > 0.5
Because we work under the assumption that the null
hypothesis is true with p = 0.5, we get the following test
statistic:
z=p–p = 0.56 - 0.5 = 3.56
pq (0.5)(0.5)
n 880
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Decision Criterion
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Traditional method:
Reject H0 if the test statistic falls within the critical region.
Fail to reject H0 if the test statistic does not fall within the critical region.
P-value method:
Reject H0 if P-value (where is the significance level, such as 0.05).
Fail to reject H0 if P-value > .
Decision Criterion
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Confidence Intervals:
Because a confidence interval estimate of a population parameter
contains the likely values of that parameter,
Reject a claim that the population parameter has a value that is not
included in the confidence interval.
Decision
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True State of Nature
The null The null
hypothesis is hypothesis is
true false
Type I error
We decide to Correct
(rejecting a true
reject the decision
null hypothesis)
Decision
null hypothesis
Type II error
We fail to Correct (rejecting a false
reject the decision null hypothesis)
null hypothesis
Controlling Type I and Type II Errors
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α is the probability of Type I error
β is the probability of Type II error
The experimenters (you and I) have the freedom to set the -level for a
particular hypothesis test. That level is called the level of significance for the
test. Changing can (and often does) affect the results of the test—whether you
reject or fail to reject H0.
• It would be wonderful if we could force both and to equal zero.
Unfortunately, these quantities have an inverse relationship. As increases,
decreases and vice versa.
• The only way to decrease both and is to increase the sample size. To make
both quantities equal zero, the sample size would have to be infinite—you would
have to sample the entire population.
