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STAM4000
Quantitative Methods
Week 8
Hypothesis testing
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| 3 ing #1 #2 #3 Describe the steps of hypothesis testing Construct hypothesis tests for one population mean Examine errors in hypothesis testing |
Week 8 Hypothesis test Learning Outcomes |
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Why does this matter?
We sometimes need to
determine if there is
significant evidence to
support a claim.
(http://4.bp.blogspot.com/-zf7S5L0XT-U/T_PX-wWXEBI/AAAAAAAADp8/DPKrX_iJJUA/s1600/1b.jpg)
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#1 Describe the steps of hypothesis testing
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#1 What is a hypothesis?
A hypothesis is an
idea,
claim or
belief
about a population,
that we want to test,
using a sample.
Photo by Jonathan Daniels on Unsplash
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#1 Steps in hypothesis testing
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#1 Step 1: Write the hypotheses
| •Expresses what we initially ASSUME to be true. |
Null hypothesis, denoted by
Ho
| •Expresses our claim into a statement that we are trying to gather enough evidence to PROVE is NOW true. |
Alternative Hypothesis,
denoted by
Ha, HA, or H1
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An individual is thought to have committed a crime and is brought before a court
of law.
In Australia, we presume the individual is innocent,
then gather evidence to try and prove they are guilty.
We have two hypotheses:
Ho: individual is innocent
Ha: individual is guilty
Objective of
testing:
gather evidence
to reject Ho and
accept Ha.
#1 Introductory example
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| 10 | Two Incorrect outcomes (errors) |
Possible outcomes in our introductory example?
Innocent
individual is freed
Guilty individual is
imprisoned
Innocent
individual is
imprisoned
Guilty individual
is freed
Four Possible outcomes
#1
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Quick quiz
This Photo by Unknown Author is licensed under CC BY
Let’s use the previous example, with the following
hypotheses:
Ho: individual is innocent
Ha: individual is guilty
Which is worse:
• an innocent individual going to prison
or
• a guilty individual going free?
#1
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#1 Step 2:Find the calculated test statistic and/or the p-value
| •Value from a formula, quantifying the difference between what is hypothesised the population and what is in the sample. |
Calculated test statistic
| •The probability of getting our sample results or more extreme, if our null hypothesis were really true. |
p-value
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#1 Step 3: Find the critical value
| •As a minimum, we need: o Relevant statistical tables or technolo o Level of significance, α o Number of tails in Ha |
Critical value
| •Area of rejection region •Probability of Type I error (later) |
Level of significance, α
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#1 Step 4: Sketch a curve
| • oOn axis of curve: ▪Insert critical value(s) ▪Label rejection region(s) ▪Insert calculated value oIn area beneath curve ▪Insert α ▪Insert p-value (if relevant) |
Ha determines the number of tails in a curve
Sketch a curve
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#1 Steps 5: Decision and Step 6: Conclusion
| i. Critical value method: compare calculated test statistic with critical value(s). or ii. p-value method: compare p-value, of calculated test statistic, with α |
Decision
If we reject Ho, then we may accept Ha.
BUT, if we fail to reject Ho, then we must RETAIN Ho;
we NEVER accept Ho.
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Conclusion
| •Use your decision to answer the original question. |
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#2 Construct hypothesis tests for one population mean
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#2 Test hypotheses about one population mean
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• Read as “alpha”.
• α is also called the significance level.
• α, and the direction in Ha, will help us define the rejection region by finding the critical
value(s).
• Usually, 1% ≤ α ≤ 10% or 0.01 ≤ α ≤ 0.10.
• α of 1% is a very strict test, as it has a very small rejection region
• α of 10% is a more lenient test, as it has a larger rejection region.
• α should be selected by the researcher before a sample is selected.
• α Is the probability of incorrectly rejecting Ho.
#2 The level of significance, α
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Denotes
position of
critical
value(s)
Ho: μ = value
Ha : μ < value 0
Ho: μ = value
Ha : μ > value
a
a
One tailed test in the left or lower tail
0
One tailed test in the right or upper tail
Two-tail test
Rejection
region(s) is
shaded
/2
0
Ho: μ = value a
Ha: μ ≠ value
Total area = 100% or 1
#2 Writing hypotheses and sketches to test μ
α is the
level of
significance
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#2 Example
i) The manager of a soda drink
company has set the bottling
machine to fill bottles to an
average volume of 250 ml.
An assembly line worker claims
the bottling machine is not
filling the bottles correctly.
The worker samples 40 bottles
and finds an average of 230 ml
per bottle.
Write the hypotheses to test the
worker’s claim.
ii) A phone
company
executive announced that the
customer call waiting time, for
their helpline, is less than, an
average of 12 minutes per call.
The company takes a random
sample of 36 calls and finds an
average wait time of 11 minutes.
Write the hypotheses to test the
executive’s announcement.
iii) The CEO of a fast-food
franchise reported that the
average weekly sales, per
franchise, is $45,000. The
marketing team believes this
can be increased and runs an
advertising campaign. The CEO
sampled the weekly sales of 10
franchises, after the campaign,
and found average sales were
$47,000.
Write the hypotheses to test
the marketing team’s belief.
This Photo by Unknown Author is licensed under CC BY-SA
This Photo by Unknown Author is licensed under
CC BY
This Photo by Unknown Author is licensed
under CC BY-NC-ND
Ho: µ = 250
Ha: µ ≠ 250
Ho: µ = 12
Ha: μ < 12
Ho : µ = 45,000
Ha: µ > 45,000
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#2
Test statistic
formula to test μ
σ is KNOWN
use Z
