Transcript Review of 10.1 and 10.2

10.1: Confidence Intervals
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Falls under the topic of “Inference.”
Inference means we are attempting to answer the
question, “How good is our answer?”
Mathematically: Confidence Interval = Estimate ±
M.O.E
Conceptually: In repeated sampling, it is the expected
percentage of intervals that would “trap” the parameter.
Or, “We arrived at this interval using a method that
yields correct results 95% of the time.”
IT DOES NOT MEAN: Our answer is 95% correct.
Our Estimate is our summary statistic, usually the
sample mean or sample proportion.
10.1: Confidence Intervals
The Margin of Error shows how accurate we believe our
guess is, based on the variability of the estimate.
 Recall from the C.L.T. that the sample mean is Normal
with a mean µ and standard deviation
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n
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 Then our Confidence Interval for µ is:
x  z*
n
z* is determined by the level of confidence we desire.
 Remember, the data MUST be from an SRS.
 Outliers can affect the confidence interval. (Why?)
 The margin of error only covers the natural variation of
the data. It DOES NOT help with nonresponse,
undercoverage, and other user errors...
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10.2: Significance Testing
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Answers the question: “How likely is this sample statistic
if we think we know the parameter?”
Ho: Null Hypothesis - The given parameter value
HA: Alternative Hypothesis - Our expectation or hope
YOU MUST ALWAYS STATE WHAT YOUR
HYPOTHESES ARE.
Test Statistic - Standardized value that measures the
deviation from our sample statistic to our parameter
value.
x 
In the case of means: Test Statistic : z  
n
P-value: Probability that if our null hypothesis
is true we
would get a sample statistic as extreme or more extreme
than what we observed.
10.2: Significance Testing
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Small P-value means that the odds of getting the sample
statistic we did are unlikely given our parameter value.
The smaller the P-value, the stronger the evidence
AGAINST the null hypothesis.
Two possibilities: Either the null hypothesis is wrong or
we got an unrepresentative sample.
We have to decide in advance how small a probability
will allow us to reject the null hypothesis.
This is called the “significance level.”
We use the Greek letter alpha to represent this.
So, if P ≤ alpha, we reject the null hypothesis
10.2: Significance Testing
Remember, “statistically significant” means we have
evidence to reject the null hypothesis.
FOUR STEPS FOR PERFORMING SIGNIFICANCE TEST
 State your null and alternative hypotheses
 Calculate the test statistic. (Be sure to show formula)
 Find the P-value and state a conclusion.
EXAMPLES OF PROPER CONCLUSIONS:
At the [alpha] level of signficance, we have sufficient evidence
to reject the null hypothesis. [P-value ≤ alpha]
At the [alpha level of signficance, we do not have sufficient
evidence to reject the null hypothesis. [P-value > alpha]
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10.2: Significance Testing
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In Chapter 10 we are often testing the null hypothesis of:
Ho: µ = µ0 where µ0 is the given parameter value
So then there are three possibilities for the P-value because
there are three place where there can be error.
P-value calculation
HA: µ > µ0; P-value is P(Z ≥ z)
[One sided, right tail]
HA: µ < µ0; P-value is P(Z ≤ z)
[One sided, left tail]
HA: µ ≠ µ0; P-value is 2P(Z ≥ |z|) [Two sided, both tails]
Note, for the last one this is the same thing as doing:
2(1- P(Z ≤ |z|))
10.2: Significance Testing
EXAMPLE:
Do middle-aged male executives have different blood pressure
than the general population? Suppose an SRS of 72
executives (aged 35 to 44) is done and the mean is 126.07. Is
this evidence that the executive blood pressures differ from
the national average (for males, 35-44) of 128? Test at the 5%
level of significance.
SOLUTION: Hypotheses: Ho: µ = 128
HA: µ ≠ 128
Test Statistic: z = -1.09
x 
Test Statistic : z 
P-value: P = 2P(Z ≥ |-1.09|)
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n
2(1 - P(Z ≤ 1.09)) = 0.2758
There is not sufficient evidence that the blood pressure of
middle-aged male executives differ from the general
population.
10.2: Significance Testing
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Remember, you can NEVER accept the null
hypothesis. We are not showing the null hypothesis to
be true. We can only “reject” or “fail to reject.”
Also, make sure you always state your conclusion in
terms of the question asked. Saying “P-value ≤ alpha”
earns zero points
Be careful about whether your test is one-sided or twosided. If you see words like “different” or “changed,”
then it’s probably a TWO-SIDED test. If you see
words like “higher,” “lower,” “better,” or “worse,”
then it’s probably a ONE-SIDED test.