Download Lecture 15 More on hypothesis testing

Hypothesis Tests
In statistics a hypothesis is a statement that
something is true.
• Selecting the population parameter being
tested (mean, proportion, variance, ect.)
• Using p-values for hypothesis tests
• Using Critical Regions for hypothesis tests
• One tailed vs. two tailed tests
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
Errors in Hypothesis Tests
A type I error is when one rejects the null
hypothesis when the null hypothesis was
true. The probability of a type I error is the
significance level  .
A type II error is when one fails to reject the
null hypothesis when the null hypothesis
was false. The probability of a type II error
is denoted  .
Example
Suppose that you are a lawyer that is trying to
establish that a company has been unfair to
minorities with regard to salary
increases. Suppose the mean salary increase
per year is 8%.
Example
Suppose that you are a lawyer that is trying to
establish that a company has been unfair to
minorities with regard to salary
increases. Suppose the mean salary increase
per year is 8%.
You set the null and alternate hypothesis:
H0:   .08
HA:   .08
Example
Suppose that you are a lawyer that is trying to
establish that a company has been unfair to
minorities with regard to salary
increases. Suppose the mean salary increase
per year is 8%.
You set the null and alternate hypothesis:
H0:   .08
HA:   .08
What is a type I error? What is a type II error?
Always express these in terms of the problem.
Example
Type I: You accuse the company of wrong
doing when they are innocent.
Type II: You let the company get away with
discrimination.
As we decrease the probability of a type I
error by changing the significance level, we
increase the chance of a type II error.
Small Samples
Up until now all the hypothesis test
examples have involved large samples
(e.g. n>30).
What a company such as BMW? They test
how many kilometers (on average) their new
cars can travel before needing to be
repaired. Each test is very expensive so
the company does not want to test 30 cars.
What a company such as BMW? They test
how many kilometers (on average) their new
cars can travel before needing to be
repaired. Each test is very expensive so
the company does not want to test 30 cars.
When testing fewer than 30 objects, you
need to use a t-statistics (the same as with
confidence intervals). Hence, we need to
find the degrees of freedom (n - 1) and use
the t-table and assume the original
population is normally distributed.
Example
Is the temperature required to damage a
computer on the average less than 110
degrees? Because of the price of testing,
twenty computers were tested to see what
minimum temperature will damage the
computer. The damaging temperature
averaged 109 degrees with a standard
deviation of 3 degrees. (Use   0.05)
Population Proportion
We have seen how to conduct hypothesis
tests for a mean and we now give some
attention to proportions.
The process is completely analogous. We
use the z-score (for large samples) and we
will need to use the standard deviation
formula for a proportion. E.g.

pq
n
Population Proportion
Example
A survey of 835 male youth showed that
401 were from single family homes. Can
we conclude that more than 45% of ll
male youth are from single family homes.
Use level of significance 0.05
Example
A survey of 835 male youth showed that
401 were from single family homes. Can
we conclude that more than 45% of ll
male youth are from single family homes.
Use level of significance 0.05
 State the research and null hypotheses
 Sketch the rejection region
 Compute the test statistic
 State your conclusion and give the pvalue.
Example
The CEO of a large electric utility claims that at
least 80 percent of his 1,000,000 customers are
very satisfied with the service they receive. To test
this claim, the local newspaper surveyed 100
customers, using simple random sampling. Among
the sampled customers, 73 percent say they are
very satisfied. The lawyers for the newspaper says
to avoid a law suit they can accuse the CEO of
misrepresentation if they are 97% certain she is
wrong. Should they print the story?
Example - Two tailed
The CEO of a large electric utility claims that
exactly 80 percent of his 1,000,000 customers are
very satisfied with the service they receive, but has
a reputation for just making up data. To test this
claim, the local newspaper surveyed 100
customers, using simple random sampling. Among
the sampled customers, 73 percent say they are
very satisfied. The lawyers for the newspaper says
to avoid a law suit they can once again accuse the
CEO of misrepresentation if they are 97% certain
she is wrong. Should they print the story? What if
the lawyer had said 95%?