Hypothesis Tests Large Sample 1- Proportion z-test

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Hypothesis Tests Large Sample 1-
Proportionz-test
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What are hypothesis tests?

• Calculations that tell us if a value occurs by
  random chance or not if it is statistically
  significant
• Is it . . .
• a random occurrence due to variation?
• a biased occurrence due to some other reason?

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Nature of hypothesis tests -
How does a murder trial work?

• First begin by supposing the effect is NOT
  present
• Next, see if data provides evidence against the
  supposition
• Example murder trial

First - assume that the person is innocent Then
must have sufficient evidence to prove guilty
Hmmmmm Hypothesis tests use the same process!
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Steps
Notice the steps are the same except we add
hypothesis statements which you will learn today

1. Define the parameter
2. Hypothesis statements
3. Assumptions
4. Calculations (Find the p-value)
5. Conclusion, in context

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Writing Hypothesis statements

• Null hypothesis is the statement being tested
  this is a statement of no effect or no
  difference
• Alternative hypothesis is the statement that we
  suspect is true

H0
Ha
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The form

• Null hypothesis
• H0 parameter hypothesized value
• Alternative hypothesis
• Ha parameter gt hypothesized value
• Ha parameter lt hypothesized value
• Ha parameter hypothesized value

Hypotheses ALWAYS refer to populations
(parameters)
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For each pair of hypotheses, indicate which are
not legitimate explain why
Must be NOT equal!
Must use same number as H0!
H0 MUST be !
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Writing Hypotheses
H0 p 0.3 Ha p lt 0.3
p 0.5 p ? 0.5
H0
Ha
H0
p 0.2 p gt 0.2
Ha
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Steps

1. Define the parameter
2. Hypothesis statements
3. Assumptions
4. Calculations (Find the p-value)
5. Conclusion, in context

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Assumptions
YEA These are the same assumptions as
confidence intervals!!

• SRS from population
• Success/Failure Condition (Large enough sample)
• np ? 10 and n(1-p) ? 10
• 10 rule the sample is less then 10 of the
  population

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Steps

1. Define the parameter
2. Hypothesis statements
3. Assumptions
4. Calculations (Find the p-value)
5. Conclusion, in context

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P-values -

• The probability that the test statistic would
  have a value as extreme or more than what is
  actually observed

In other words . . . is it far out in the tails
of the distribution?
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Level of significance -

• Is the amount of evidence necessary before we
  begin to doubt that the null hypothesis is true
• Is the probability that we will reject the null
  hypothesis, assuming that it is true
• Denoted by a
• Can be any value
• Usual values 0.1, 0.05, 0.01
• Most common is 0.05 (default value)

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Statistically significant

• The p-value is as small or smaller than the level
  of significance (a)
• If p gt a, fail to reject the null hypothesis at
  the a level.
• If p lt a, reject the null hypothesis at the a
  level.

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Facts about p-values

• ALWAYS make decision about the null hypothesis!
• Large p-values show support for the null
  hypothesis, but never that it is true!
• Small p-values show support that the null is not
  true.
• Double the p-value for two-tail () tests
• Never accept the null hypothesis!

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Never accept the null hypothesis!
Never accept the null hypothesis!
Never accept the null hypothesis!
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At an a level of .05, would you reject or fail to
reject H0 for the given p-values?

• .03
• .15
• .45
• .023

Reject
Fail to reject
Fail to reject
Reject
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Formula for hypothesis test
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Calculating p-values

• One sided test (lt)

P-value P (z lt calculated value)

• One sided test (gt)

P-value P (z gt calculated value)

• Two sided test (?)

P-value 2P (z lt calculated value)
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Steps

1. Define the parameter
2. Hypothesis statements
3. Assumptions
4. Calculations (Find the p-value)
5. Conclusion, in context

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Writing Conclusions

• Decision A statement of the decision being made
  (reject or fail to reject H0) why (linkage)
• Conclusion A statement of the results in
  context. (state in terms of Ha)

AND
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Decision
Since the p-value ? ?, I reject the null
hypothesis at the ? level.
or
Since the p-value gt ?, I fail to reject the null
hypothesis at the ? level.
A statement about Ha in context (words)!
Conclusion
There is enough evidence to conclude that the
true proportion of ...
or
There is not enough evidence to conclude that the
true proportion of ...
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A company is willing to renew its advertising
contract with a local radio station only if the
station can prove that more than 20 of the
residents of the city have heard the ad and
recognize the companys product. The radio
station conducts a random sample of 400 people
and finds that 90 have heard the ad and recognize
the product. Is this sufficient evidence
for the company to renew its
contract?
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Parameter and Hypotheses
p the true proportion of people who heard the ad
H0 p .2 Ha p gt .2
Use the parameter in the null hypothesis to check
assumptions!
Assumptions (Conditions)

1. The sample must be random which is stated in the
   problem.
2. The sample should be large. Since np 400(.2)
   80 gt10 and n(1-p) 400(.8) 320 gt10, the
   sample is large enough.
3. The sample must be less then 10 of the
   population. The population should be at least
   4000 people which I will assume it to be.

Since the conditions are met, a z-test for
proportions is appropriate.
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Calculations
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Since the p-value gt ?, I fail to reject the null
hypothesis at the .05 level.
Decision
Conclusion
There is not enough evidence to conclude that the
true proportion of people who heard the ad is
greater than .2