Transcript Inferences about Means

Chapter 23
Inferences About Means
Sampling Distribution for the Mean
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Now that we know how to create confidence intervals and
test hypotheses about proportions, it’d be nice to be able
to do the same for means.
Just as we did before, we will base both our confidence
interval and our hypothesis test on the sampling
distribution model.
The Central Limit Theorem told us that the sampling
distribution model for means is Normal with mean μ and
standard deviation
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SD  y  
n
All we need is a random sample of quantitative data, but
the true population standard deviation, σ is unknown, so
we use standard error:
s
SE  y  
n
Example
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A nutrition laboratory tests 61 “reduced sodium” hot dogs,
finding that the mean sodium content is 320 mg, with a
standard deviation of 37 mg.
Find a 95% confidence interval for the mean sodium
content of this brand of hot dog.
Answer: the mean is between 310.4 and 329.6 mg
Student’s t distribution
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Unfortunately, the shape of the sampling model changes—
the model is no longer Normal. So, what is the sampling
model?
The sampling model found has been known as Student’s t.
The Student’s t-models form a whole family of related
distributions that depend on a parameter known as
degrees of freedom.
 We often denote degrees of freedom as df, and the
model as tdf.
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As the degrees of freedom increase, the t-models look
more and more like the Normal.
In fact, the t-model with infinite degrees of freedom is
exactly Normal.
Assumptions and Conditions
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Independence Assumption:
 Randomization Condition
 10% Condition
 Nearly normal condition
A Confidence Interval for Means
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One-sample t-interval for the mean
When the conditions are met, we are ready to find the
confidence interval for the population mean, μ.
The standardized sample mean t  y  
SE  y 
follows a Student’s t-model with n – 1 degrees of freedom.
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 The confidence interval is
y  t  SE y
n1
where the standard error of the mean is
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SE  y  
s
n
The critical value tn*1 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 Test for the Mean
One-sample t-test for the mean
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The assumptions and conditions for the one-sample t-test
for the mean are the same as for the one-sample t-interval.
We test the hypothesis H0:  = 0 using the statistic
y  0
tn1 
SE  y 
s
SE  y  
n
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The standard error of the sample mean is
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When the conditions are met and the null hypothesis is true,
this statistic follows a Student’s t-model with n – 1 df. We
use that model to obtain a P-value.
Finding t-Values By Hand
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The Student’s t-model
is different for each
value of degrees of
freedom.
Because of this,
Statistics books
usually have one table
of t-model critical
values for selected
confidence levels.
Example
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One of the most important factors of auto safety
is the weight of the vehicle. Insurance companies
are interested in knowing the average weight of
cars currently licensed. They believe it is 3000
pounds. To see if this estimate is correct, the
company checked a random sample of 91 cars.
For that group the mean weight was 2866
pounds, with a standard deviation of 531.5
pounds. Is this strong evidence that the mean
weight of all cars is not 3000 pounds?
Homework Assignments
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Page 609 – 613
Problem # 1, 7, 19, 33, 37, 43 a,