STAB22H3 Chapter Notes - Chapter 19-20: Sample Size Determination, Statistical Hypothesis Testing, Standard Deviation

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8 Feb 2016
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STAB22: TEXTBOOK NOTES
PART IV: CHAPTERS 19-20
Naomi
Alphonsus
Example: Polls and margin of error
June ’08, survey of 100 Cdns found that 68% of Canadians support gay marriage. Researchers reported margin o
error to be +/- 3.1%. Question: Standard among pollsters is to use the 95% confidence level unless otherwise
stated. Given that, what do they mean by claiming a margin error of +/- 3.1% in this context?
o If this polling were done repeatedly, 95% of all random samples would yield estimates that come within
3.1 percentage points (0.031) of the true percentage (proportion) of all Cdns who support gay marriage
Example: Finding the margin of error
A june 2008 poll of 1000 canadians reported a margin of error of +/- 3.1%. The worst case margin of error,
generally among pollsters is passed on =0.5. How did Strategic Counsel calculate their margin of error?
o Assuming p=0.5, for random samples of size = 1000
SD()=
= 
 = 0.0158
For a 95% confidence level, ME= 2(0.0158) = 0.032
The margin of error is just a bit over +/- 3%
CHAPTER 19: CONFIDENCE INTERVALS AND PROPORTIONS
A confidence Interval
Standar error: calculated standard deviation of a sampling distribution
For a sample proportion, , the standard error is:
o SE ()=
Ex. sample proportion of sea fans affected by the disease aspergillosis
A confidence interval is an estimate +/- margin of error
What does 95% condidence really mean/
95% of samples of this size will produce confidence intervals that capture the true proportions
o The true proportion lies within our interval
o Uncertainity is whether the particular sample is one of the successful ones or one of the 5% that fail to
produce an interval that captures the true value
Margin of Error: Certainty versus Precision
The extent of the inteveal on either side of is called the margin o error (ME). Confidence intervals look like this:
Estimate +/- ME
The more confident we want to be, the larger the margin of error must be.
The most commonly chosen confidence levels are 90%, 95% and 99%
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Naomi Alphonsus Stab22 textbook notes 2
Example: finding the margin error (take 2)
Recap: 1000 Canadians polled, 68% of Canadians support gay marriage. 95% level of confidence; margin of error
= +/- 3.1%
Question: Using the critical value of z and the standard error based on the observations proportion, what would
be the margin of error for a 90% level of confidence? What’s good and bad about this change in level of
confidence?
o N= 1000  
o SD()=
= 
 =0.0148
o For a 90% confidence level, z*=1.645
ME= 1.645 (0.0148)=0.024
The margin of error is closer to +/- 2% producing a narrower interval. Makes for a more precise estimate of
voter belief but provides less certainty that the interval actually contains the true proportion of voters
supporting gay marriage
Critical Values
To change the confidence level, we’d need to change the number of SEs so that the size of the margin of error corresponds
to the new level called critical value.
Critical value: the number of SE’s to move away from the mean of the sampling distribution to correspond to the specified
level of confidence. It is denoted by z* and usually found from a table or with technology
Precise critical value for a 95% confidence interval is z*=1.96
Assumptions and Conditions
Assumptions made are not just about how our data look but about how representative they are
Independence assumption
Independence assumption: we wonder whether there is any reason to believe that the data values somehow
affect each other. It depends on knowledge of the situation, isn’t tested to be plausible by looking at the data
Randomization condition: were the data sampled at random or generated from a properly randomized
experiment? Proper randomization can ensure independence. SE formula not valid I multi-stage or other complex
sampling design was used.
10% condition: when we sample from small populations, the probability of success may be different for the last
few individuals we draw than it was for the first few. If the sample exceeds 10% of the population, the probability
of success changes so much during the sampling that our Normal model may no longer be appropriate. If less than
10@ of the population is sampled, the effect on independence is negligible.
Sample size Assumption
This is based on the ventral limit theorem. The Sample size assumption addresses the question of whether the sample is
large enough to make the sampling model for the sample proportions approx normal
Success/failure condition: we must expect at least 10 “successes” and at least 10 “failures”
Choosing your sample Size
Suppose a candidate is planning a poll and wants to estimate voter support within 3 percentage points with95 %
confidence. How large a sample does she need?
ME= z* 
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