Download Section 7.1

More Inferences About Means
Student’s t distribution and
sample standard deviation, s
Reconsider inferences about m,
the population mean

When we make a CI or calculate the
test statistic for hypotheses involving m
we use s, the standard deviation of the
population from which we’re sampling.
– The standard deviation of the sample
mean is s/sqrt(n).
– However, we often don’t know s!
Using s, the sample standard
deviation, to estimate s

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If we don’t know s, we need to estimate
it from the sample.
We use s as an estimate of s.
– s is discussed in Ch. 1 (see p.48).
–
1
2
xi  x 
s

n 1
Increased Uncertainty

We’d like to make inference about m, the
unknown population mean.
– We use the sample mean as an estimator of m.
– Now, we also use s as an estimate of s.

This results in increased uncertainty about
the sample mean we’re likely to obtain.
– What distribution describes this uncertainty?
– Student’s t distribution.
Student’s t distribution

The t distribution…
– Is similar to the normal distribution.
– Has heavier tails than the normal distribution.
– Exists with varying degrees of freedom (d.f.).
• When degrees of freedom are low, tails are heaviest.
• As degrees of freedom increase without bound, the t
distribution converges to the normal distribution.
T statistic

The t-statistic, t, is used for inference of the
mean of a population, when s is unknown.
xm
t
s n
– This test statistic has a t distribution with n  1
degrees of freedom.
– The margin of error, m, for a CI is
mt
*
s
n
where t* is the appropriate value from the t
distribution with n  1 degrees of freedom.
Assumptions

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When we use the t distribution, we assume
the population from which we’re sampling is
normally distributed.
However, hypothesis tests and CIs using the t
distribution are “robust” inference techniques.
– They can often be used for even very non-normal
populations if n  40.
– If n <15, we must be sure that population
distribution is very close to normal.
Example: Housing Prices

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A real estate agency in a big city wants to test
whether the mean home price exceeds
$132,000 (using a = 0.10).
25 recent sales are randomly chosen and
these have an average sales price of
$148,000 and s = $62,000.
Perform the t-test.
– What assumptions are needed?
– What hypothesis is supported?
Example: Bottling Factory
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A factory fills 20 oz. bottles with soda.
Assume the amount of soda in a bottle has a
normal distribution.
A random sample of bottles was taken from
the factory line (data in
P:\Data\Math\Radmacher\bottles.mtw).
– Is there evidence (at a = 0.05) to make us think
that the mean filling level is not 20 oz.?

You want to rent an unfurnished one bedroom
apartment. You take a random sample of 10
apartments advertised in the Mount Vernon
News and record the rental rates. Here are
the rents (in $ per month):
500, 650, 600, 505, 450, 550, 515, 495, 650, 395
– Find a 95% CI for the mean monthly rent for
unfurnished one bedroom apartments in the
community.
– Do these data give good reason to believe that the
mean rent of all such apartments is greater than
$500 per month?