Download Chapter 23 - Huber Heights City Schools

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?
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9.7
10.0
10.1
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10.0
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10.1
9.9
9.6
10.2
9.8
10.0
9.9
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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?