AP Statistics: Chapter 23

```Inferences About Means
this new program, a random sample of 45 eighth-grade students
was selected. These students participated in the new reading
program for one semester and then took a standard readingcomprehension examination. The mean test score for the
population of students who had taken this test in the past was 76
with a standard deviation of  = 8. The sample results for the 45
students provided a mean of 79. Is there significant evidence at
the .05 level of significance that scores have improved with the
new program?
2. The pain reliever currently used in a hospital is known to bring
relief to patients in a mean time of 3.9 minutes with a standard
deviation of 1.14 minutes. To compare a new pain reliever with
the current one, the new drug is administered to a random
sample of 40 patients. The mean time to provide relief for the
sample of patients is 3.5 minutes. Do the data provide sufficient
evidence to conclude that the new drug was effective in reducing
the mean time until a patient receives relief from pain? Use a
.01 level of significance.
If our data come from a simple
random sample (SRS) and the
sample size is sufficiently large,
then we know that the sampling
distribution of the sample means is
approximately normal with mean

μ and standard deviation n .
PROBLEM:
If &Ccedil; is unknown, then we cannot calculate the
standard deviation for the sampling model. 
We must estimate the value of &Ccedil; 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 &Ccedil;.
Then the standard error of the sample mean ˜ is
s
.
n
In order to standardize ˜, we subtract its mean and
divide by its standard deviation.
x
z

has the normal distribution N( 0, 1).
n
PROBLEM:
If we replace &Ccedil; 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 a standardized value. Like z, t tells us how
many standardized units ˜ is from the mean
&Atilde;.
When we describe a t-distribution we must
identify its degrees of freedom because
there is a different t-statistic for each
sample size. The degrees of freedom for the
one-sample t statistic is n – 1 .
The t distribution is symmetric about zero and is
bell-shaped, but there is more variation so the
Normal Distribution
t- Distribution
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 

 n
*
df
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.
The t procedures are strongly influenced by
outliers . Always check the data first! If
there are outliers and the sample size is
small , the results will not be reliable.
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 customers?
 Use
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:
 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, 336, 311,
253, 270, 326, 311, and 288.
 Find a 90% confidence interval for the mean
lifespan of this type of battery.
 Use 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?
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