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Chapter

STAB22-C20.docx


Department
Statistics
Course Code
STAB22H3
Professor
Ken Butler

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STAB22 - C20
TESTING HYPOTHESES ABOUT PROPORTIONS
[2]
(Ex-IGNOTS)
- huge pieces of metal that are used as structural parts for cars and planes
- problem: if they crack while they are manufactured, they must be remade, or
recycled, but recycling costs too much $
- before changes were made to manufacturing, only 80% free of cracks
- after changes, 400 ignots were sent for manufacturing and out of which 83% free of
cracks
- has the cracking rate really reduced, or was this by chance?
p531
[1]
(Ex-IGNOTS)
- treat 400 ignots as random sample
- recall: each random sample has diff. prop. of cracked ignots (b/c there is
variation from one sample to another)
- so we want to know:
- is this 17% due to sampling variability, or is it strong enough evidence that the
true cracking rate is now really below 20%?
[2]
(Ex-IGNOTS)
- the above qn, and ex's below are answered by testing
hypotheses
about models:

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- ex. has the PM's approval rating changed since the last yr?
- ex. has teenager smoking been reduced in past 3 yrs?
- ex. is global T decr'ing?
HYPOTHESES
[1]
Hypotheses = working models that are adopted temporarily
- ie. used temporarily
(Ex-IGNOTS)
- to test whether changes made to manufacturing process really did improve cracking
rate, assume that:
- it made no difference, & so any sort of improvement is really just random
fluctuation (sampling error)
= null hypothesis in this context
- states that prop. of cracks really is still 20%
- ie. p = 0.20
NULL HYPOTHESIS - assumption that there is no real difference between the
groups/treatments in comparison, and that if there is any sort, it is merely due to
random fluctuation (Sampling error)
[3]
NULL HYPOTHESIS
- denoted as H0
- specfies popn model parameter of interest
- proposes val. for that paramter
- typically written in this form:

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parameter
=
hypothesized value
- specific hypothesized val. for parameter, which is used to compare against
observed statistic val.
[4]
NULL HYPOTHESIS (Ex-INGOTS)
H0 :
p
= 0.20
- what val. it takes on is typically obvious from Who and What of data
[5]
ALTERNATIVE HYPOTHESIS
- denoted as HA
- contains val's of parameter that we consider plausible IF we reject null hypothesis
ALTERNATIVE HYPOTHESIS (Ex-INGOTS)
- H0 is that:
p
= 0.20
- HA is p < 0.20
- we are interested in reducing the cracking rate
- this is sensible alternative other than making
p
equal 0.20
[6]
- what would make us think that cracking rate really went lower than 20%?
- if we observed cracking rate was significantly lower than 20%, then this is
likely convincing
- ex. only 3/400 ingots crack (0.75% rate)
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