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Estimation of µ
• The distribution of sample means (the
sampling distribution) is a (nearly)
normal distribution.
• This distribution has a mean and a
variance, or error. The mean is , and
the measure of variance that is used is
called the standard error of the mean,
X
and is symbolized
• Formula for standard error of the mean
is sigma divided by the square root of

sample size, or
n
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CLT
• As sample size increases to equal to or greater than 10 , the
sampling distribution approaches normality. This
phenomenon is referred to as the Central Limit Theorem.
• Definition of CLT: For any populations with mean µ and
standard deviation , the distribution of sample means for
sample size n will approach a normal distribution with a
mean of µand a standard error of /[sq.rt(n)] as n
approaches infinity.
• The sampling distribution with n≥30 is normal, regardless of
the shape of the original population.
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Law of Large Numbers
• The law of large numbers states that the larger
the sample size (n), the more probable it is that
the sample mean will be close to the population
mean.
• Large samples are reliable, due to the lack of
variability in the sampling distribution.
• The estimate of the standard error of the mean:
s
n
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HYPOTHESIS TESTING
• Always test Ho, the null hypothesis.
• The null hypothesis states that
takes on a specified value.
• Always include an alternative
hypothesis, H1.
• The alternative hypothesis states that
 takes on a value other than that
specified by Ho.
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Formalization
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The set-ups:
Ho: µ=V, where V is some specified value
H1: µ≠V
Ho: µ=V, where V is some specified value
H1: µ<V
Ho: µ=V, where V is some specified value
H1: µ>V
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Test of the null hypothesis
z
X
X
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Example time

They say everything is bigger in Texas.
(pause for hilarious laughter from the class.)
M,
the naïve lad from the Savannah, is told that the
mosquitoes are so large that they can put dents in car
fenders. Although naïve, M is not stupid, so he knows
hyperbole when he hears it. The Texans tell M that the
average Texas “skeeter” weighs in at a remarkable 3000
mg. M is both impressed and terrified, and given his
scientific tendencies, decides to test this claim. So, he
goes around the state, obtaining a random sampling of
225 of the critters. One thing that M does know for
certain from reading his entomology books is that the
population standard deviation for the weight of Texas
mosquitoes is 75mg.
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Example (cont.)
M’s sample yields an average weight of
3020mg. So, the question is, do we reject
or fail to reject the claim made by these
crazy Texans?
 Ho: µ=3000mg
 H1: µ≠3000mg

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cont.

Ho: µ=3000mg

H1: µ≠3000mg

75mg; n=225;

 X =75/

Remember
=3020mg
X
225= 5
z
X
X
?
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Test of the null hypothesis (cont.)

Reject the null hypothesis when z indicates
that the sample mean is an extreme value.
How extreme? When the probability of
sampling a mean as extreme or more
extreme than the sample mean occurs 5% of
the time or less.

When to reject the null hypothesis? If your zscore indicates a sample mean that occurs
with a frequency of 5% or less.
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Decision Rule

The z-score that we calculate is referred to as
zcalc. The z-score based on the alpha level
(what the hell is that?) is referred to as zcrit.

The decision rule is straightforward. When
abs. value of zcalc equals or exceeds abs.
value of zcrit,, reject the null hypothesis.
When zcalc does not exceed, conclude that
your results are inconclusive --you “fail to
reject” the null hypothesis.
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Significance levels and probability values
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This 5% value is called the significance level or  level.
Significance (or  level): The probability level that must
be equaled or exceeded in order to reject the null.
Usually, .05; sometimes .01
p-value: Probability of observing a mean as extreme or
more extreme, assuming a true null hypothesis
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e.g., If a sample mean has an associated p-value of .20, then
assuming Ho to be true, the likelihood of sampling a mean this
extreme (or more extreme) is .20 (or 20%)
More about p-values.

Let’s relate p-values to the “big picture” (normal curve).
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continued

Another way of saying this is that 20% of
the time, you will draw samples with
means that are this extreme (far from the
mean specified by Ho) or more extreme.
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Variations on this example

We can change the sample mean (make it
larger or smaller).
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We can alter the sample size (again, increasing
or decreasing it).
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This will affect the p-value
This, too, will affect the p-value by affecting the
standard error of the mean.
We can alter our claim (Change the value of the
null hypothesis).

This will also affect the p-value!
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One- vs. two-tailed hypothesis
tests
• When the alternative hypothesis includes a ">" or
"<", one-tailed test (because the focus is on only
one tail of the distribution.)
• When the alternative hypothesis includes a "≠",
two-tailed test (because both tails are in play.)
• All things being equal, harder to reject the null
hypothesis using a two-tailed test (because z-crit is
larger than it is under one-tailed conditions).
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A few examples

Crazy Johansen claims to have read that the average Odessan loves
football so much that each residents spends an average of 26 hours/week
talking about football. You send your crack team out to west Texas to
investigate. After spying on a random sample of 36 residents, you find the
following:
 Mean: 38 hrs/week
 : 10 hrs
/
z = (38 - 26) 10/sqrt(36)
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A few examples

Larry the Loon claims that the average Austin house is constructed with
1,035 nails hammered into the wood framing. You and your crack team
decide to investigate. Under the cover of darkness, you disassemble 16
houses, and after hiring a good bail bondsman to free your crack team,
you report the following:
 Mean: 1,300 nails/house
 : 400 nails
/
z = (1,300 - 1,035) 400/sqrt(16)
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