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Chapter 23
INFERENCE ABOUT MEANS
CLT!!
 If our data come from a simple random sample
(SRS) and the sample size is sufficiently large, then
we know the sampling distribution of the sample
means is approximately normal with mean µ and

standard deviation .
n
Problem
 If σ is unknown, then we cannot calculate the
standard deviation for the sampling model. 
 We must estimate the value of σ in order to use the
methods of inference that we have learned.
Solution
 We will use s (the standard deviation of the sample)
to estimate σ.
 Then the standard error of the sample mean
x is
s
.
n
 In order to standardize
x , we subtract its mean and
divide by the standard deviation.
z
x

 __________has
the normal distribution N(0,1).
n
Problem
 If we replace σ with s, then the statistic has more
variation and no longer has a normal distribution so
we cannot call it z. It has a new distribution called
the t distribution.
 t is the standard value. Like z, t tells us how many
standardized units is from the mean µ.
 When we describe a t-distribution we must identify
its degrees of freedom because there is a different
statistic for each sample size. The degrees of
freedom for the one-sample t statistic is n – 1.
x
 The t distribution is symmetric about zero and is
bell-shaped, but there is more variation so the spread
is greater.
 As the degrees of freedom increase, the t distribution
gets closer to the Normal distribution, since s gets
closer to σ.
 We can construct a confidence interval using the t
distribution in the same way we constructed
confidence intervals for the z distribution.
*  s 
x  t df 

 n
 Remember, the t Table uses the area to the right of
t*.
 One sample t procedures are exactly correct only
when the population is Normal. It must be
reasonable to assume that the population is
approximately normal in order to justify the use of t
procedures.
When to use t procedures:
 If the sample size is less than 15, only use t
procedures if the data are close to Normal.
 If the sample size is at least 15 but less than 40 only
use t procedures if the data is unimodal and
reasonably symmetric.
 If the sample size is at least 40, you may use t
procedures, even if the data is skewed.
Example
 A coffee vending machine dispenses coffee into a
paper cup. You’re supposed to get 10 ounces of
coffee, but the amount varies slightly from cup to
cup. Here are the amounts measured in a random
sample of 20 cups. Is there evidence that the
machine is shortchanging the customer?
9.9
9.7
10.0
10.1
9.9
9.6
9.8
9.8
10.0
9.5
9.7
10.1
9.9
9.6
10.2
9.8
10.0
9.9
9.5
9.9
PHANTOMS!!
Example 2
 A company has set a goal of developing a battery that
lasts five hours (300 minutes) in continuous use. In
a first test of these batteries, the following lifespans
(in minutes) were measured: 321, 295, 332, 351, 311,
253, 270, 326, 311, and 288.
 Find a 90% confidence interval for the mean lifespan
of this type of battery.
PANIC!!!
 If we wish to conduct another trial, how many
batteries must we test to be 95% sure of estimating
the mean lifespan to within 15 minutes?
To within 5 minutes?