Transcript Chapter 23: Inferences About Means Confidence Intervals & Hypotheses About Means  To create confidence intervals and test hypotheses about means:    Base both on the sampling.

Chapter 23:
Inferences About Means
Confidence Intervals & Hypotheses
About Means
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To create confidence intervals and test
hypotheses about means:
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Base both on the sampling model
CLT tells us that the sampling model is
Normal
Standard Error is just the estimated
standard deviation of the sampling model
Gosset’s t
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Gosset worked as a quality control
engineer.
He noticed that with small sample
size, his tests for quality weren’t
quite right.
When he used the estimated
standard error, the shape of the
sampling model changed; he called
the new model a t-distribution.
Student’s t-models form a whole
family of related distributions that
depend on a parameter known as
degrees of freedom (df).
A Sampling Distribution for Means
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When the conditions are
met, the standardized
sample mean,
t
y
 
SE y
follows a Student’s t-model
with n – 1 degrees of
freedom.
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We estimate the
standard error with
 
SE y 
s
n
Gosset’s Model
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When Gosset corrected the model for
the extra uncertainty, the margin of
error got bigger.
When you use Gosset’s model instead
of the Normal model, your confidence
interval will be slightly wider and your
P-values slightly larger.
“To t or not to t?”
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If you know
use z (very
rare!).
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Whenever you use s to
estimate  , use t.
Student’s t-models are
unimodal, symmetric
and bell shaped.
Assumptions and Conditions
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Independence Assumption
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Randomization condition
10% condition
Normal Population Assumption
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Nearly Normal condition
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The data come from a distribution that is
unimodal and symmetric.
Check by making a histogram or Normal
probability plot.
One-sample t-interval
When the conditions are met, find the confidence level
for the population mean, . Since the standard error of the mean is

SE y 
t * n 1
s

, the interval y  t * n 1  SE y . The critical value
n
depends on the particular confidence level, C , that you specify
and on the number of degrees of freedom, n  1, which we get from
the sample size.
A One-Sample t-Interval for the Mean
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Identify the parameter:
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Find a 90% confidence interval for the mean
speed of vehicles driving on Triphammer Road.
Look at the data:
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Enter data into L1.
A One-Sample t-Interval for the Mean
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Check the conditions:
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Randomization: we have a convenience sample,
but we have reason to believe it is
representative.
10%: the cars observed were fewer than 10%
of al cars traveling on Triphammer Road.
Nearly Normal Condition: The histogram is
unimodal and symmetric
A One-Sample t-Interval for the Mean
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State the sampling distribution model for
the statistic.
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Under these conditions the sampling
distribution of the mean can be modeled by
Student’s t-model with 22 degrees of freedom:
n  1  23  1  22
Choose your method.
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We will use a one-sample t-interval for the
mean.
A One-Sample t-Interval for the Mean
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Construct the
confidence interval
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Under STAT TESTS
choose Tinterval
We know
n  23 cars
y  31.0 mph
s  4.25 mph
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Margin of Error:
 
ME  t *22  SE y
 1.717  .0886 
 1.521 mph
A One-Sample t-Interval for the Mean
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Interpretation:
 We are 90% confident that the true mean speed
of all vehicles on Triphammer Road is between
29.5 and 32.5 miles per hour.
 Caution: this was not a random sample of
vehicles. It was a convenience sample taken at
one time of the day. The drivers could possibly
have seen the police device and may have
slowed down. We are reluctant to extend our
inference to other situations.
A One-Sample t-Test for the Mean
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State the hypotheses:
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We want to know whether the mean speed of
vehicles on Triphammer Road exceeds the
posted speed limit of 30 mph.
State the null hypothesis:
HO : Mean speed,   30 mph
H A : Mean speed,   30 mph
A One-Sample t-Test for the Mean
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The histogram:
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Check the conditions:
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Randomization: we have a
convenience sample, but we
have reason to believe it is
representative.
10%: the cars observed were
fewer than 10% of al cars
traveling on Triphammer Road.
Nearly Normal Condition: The
histogram is unimodal and
symmetric.
A One-Sample t-Test for the Mean
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State the sampling distribution model for the
statistic.
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Under these conditions the sampling distribution
of the mean can be modeled by Student’s tmodel with 22 degrees of freedom:
n  1  23  1  22
Choose your method.
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We will use a one-sample t-test for the mean.
A One-Sample t-Test for the Mean
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STAT TESTS T-Test
Calculate:
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STAT TESTS T-Test
Draw:
A One-Sample t-Test for the Mean
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Conclusion:
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Link the P-value to your decision about the null
hypothesis and state your conclusion in context.
The P-value of 0.126 says that if the true mean speed of
vehicles on Triphammer Road were 30 mph, samples of
23 vehicles can expected to have an observed mean of
at least 31.0 mph 12.6% of the time. That P-value is
not small enough for us to reject the hypothesis that the
true mean is 30 mph at any alpha level. We conclude
that there is not enough evidence to say that the
average speed is too high.
Intervals and Tests
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Confidence intervals and significance tests are
built from the same calculations.
The confidence level contains all the null
hypothesis values you can’t reject.
A level C confidence interval contains all of the
possible null hypothesis values that would be
retained by a two-sided hypothesis test at -level
1 – C.
When the hypothesis is one-sided, the
corresponding -level is (1 – C)/2.
Sample Size
Before collecting data, it is a good idea to know whether the
sample size is large enough to give you a good chance of
being able to tell you what you want to know.
An example: the movie download p. 456
ME  8 min, SD  10 min, 95% confidence interval
8  1.96
10
n  2.45 n  6.0025
n
Use  6  1  5 degrees of freedom to substitute an appropriate
t * value.
8  2.571
10
n  3.214 n  10.33
n
Round up, so n  11 movies
CAUTION!!
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Beware multimodality.
 Look for the possibility that the data come from two groups.
 If so, separate the groups and analyze each group separately.
Beware skewed data.
Set outliers aside.
 Report on these values separately.
 Conduct an analysis of non-outlying points, along with a separate
discussion of outliers.
Watch out for bias.
 Think about possible sources of bias in your measurements.
Make sure data are independent.
Make sure that data are from an appropriately randomized sample.