Download Chapter 11.1 t

Section 11.1
Inference for the Mean
of a Population
AP Statistics
March 10, 2009
AP Statistics, Section 8.2.1
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AP Statistics, Section 8.2.1
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The t statistic

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The t statistic is used when
we don’t know the
standard deviation of the
population, and instead we
use the sample distribution
as an estimation.
The t statistic has n-1
degrees of freedom (df).
x 
t
s/ n
The t statistic

In statistical tests of
significance, we still have
H0 and Ha.

We need to provide the
mu in the calculation of
the t statistic.

Looking at the t table is
fundamentally different
than the z table.
x 
t
s/ n
AP Statistics, Section 8.2.1
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AP Statistics, Section 8.2.1
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The t statistic




The t statistic is bigger than the z statistic.
We say that t distribution is a more conservative
distribution.
There is more area in the tails.
The t statistic has n-1 degrees of freedom.
AP Statistics, Section 8.2.1
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AP Statistics, Section 8.2.1
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Example

Construct and interpret a 95% confidence
interval for the mean amount of (NOX)
emitted by light duty engines of this type.

Step 1: Parameter: The population of interest is
all light duty engines of this type. We want to
estimate  , the mean amount of the pollutant
NOX emitted, for all of these engines.

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Step 2: Conditions: Since we do not know σ, we
should construct a one-sample t-interval for the
mean NOX level  if the conditions are satisfied.
SRS: The data comes from a random sample of 46
engines from the population of all light-duty engines
of this type. We are not told the data comes from a
SRS so we proceed with caution.
Normality: Is the population distribution of NOX
emissions Normal? We do not know from the
problem statement. Let’s examine the sample data.
Independence: We must assume that there are at
least (10)(46) = 460 light-duty engines of this type
since we are sampling without replacement.

Step 3: Calculation: Check that the mean NOX
emission reading for the 46 light-duty engines in
our sample is x-bar = 1.329 grams per mile.

Using a graphing calculator we calculate a
t-interval, inputting the degrees of freedom at 45.

(Using the table we would have to use a degree
of freedom of 40 since 45 is not an option)
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Step 4: Interpretation: We are 95% confident
that the true mean level of nitrogen oxides
emitted by this type of light-duty engine is
between 1.185 and 1.473 grams/mile.
Example: Mr. Young Mopping

Let’s suppose that Mr. Young
has been told that he should
mop by 25 after 1.

We collect 12 samples with
an average 27.58 minutes
after 1 p.m. with a standard
deviation of 3.848 minutes.

Is this evidence that his true
mean is after 1:25?
x 
t
s/ n
Step 1: Mr. Young Mopping

Population of interest:
 Mr.

Young’s mopping
Parameter of interest:
 average
time of arrival
during mopping
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Hypothesis
 H0:
µ=25
 Ha: µ>25
x 
t
s/ n
Step 2: Mr. Young Mopping
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We are using 1 sample t-test?
SRS?
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Normality?
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No. Proceed with caution.
Big sample size (> 40)
Sample is somewhat normal
because the sample distribution
is single peaked, no obvious
outliers.
Population size is at least 10
times the sample size?

We assume that Mr. Young has
done a lot of mopping
x 
t
s/ n
Step 3: Mr. Young Mopping

Calculate the test statistic, and calculate the p-value.
27.58  25
t
3.848 / 12
 2.322
P(t  2.322) is between .025 and .02
AP Statistics, Section 8.2.1
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Exercises
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11.1-11.19 odd, 11.21-11.35 odd
AP Statistics, Section 11.1
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