Chapter 23: Inferences about Means
•
The one exception is when we have a model that gives us a value for sigma.
If we know sigma, it would be foolish not to use this information, and if we
don’t have to estimate s, then we can use the Normal model.
•
As the degrees of freedom increase,
the tdistributions become more and more like the normal model
•
For confidence intervals: Don’t say:
o
90% of all the vehicles on Triphammer Road . . . because the confidence interval is about mean speed.
o
We are 90% confident that a randomly selected vehicle will have a mean speed . . . because this doesn’t refer to means as well.
o
The mean speed of vehicles is 31.0 mph 90% of the time because it implies totality.
o
90% of all samples . . . because we can not assume to know the mean of all samples, which will vary from sample to sample.
1.
One Sample TInterval
:
2.
One Sample T Test
:
= H
0
: Mean speed (µ) = 30
= H
A
: Mean speed (µ) > 30
Randomization Condition, Nearly Normal Condition = the histogram of the speeds is unimodal and reasonably symmetric (fails if two or more modes).
This is close
enough to normal for our purposes.
The P value is the probability of observing a sample mean as large as 31.0 mph or larger if the true mean were 30 mph.
= n = 23 cars, so 23 – 1 = 22 degrees of freedom
= SE(y
bar
) = s / sqrt(n)
= t = (y
bar
 µ
0
) / SE(y
bar
)
P = 0.136, so the P value says that if the true mean speed of vehicles on Triphammer Road
were 30 mph, samples of 23 vehicles can be expected to have an observed
mean of at least 31.0 mph 13.6% of time.
The P value is not small enough for us to reject the hypothesis that the true mean is 30 mph at any reasonable alpha level.
We conclude that there is not enough evidence to say that the average speed is too high.
•
When the hypothesis is one sided, the corresponding alpha level is 1 – C / 2
Chapter 24: Comparing Means
The statistic of interest is the difference between the two observed means y
bar
1 – y
bar
2.
To find the standard deviation, we
add the variances and then take their
square root
.
The confidence interval that is used in this case is called a
two sample tinterval
and the corresponding hypothesis test is called a
two sample ttest
.
These formulas are almost exactly the same as those in Chapter 22 for comparing two proportions with the exception that we use Student’s tmodel instead of the
Normal Model.
a.
Independence assumption, normal population assumption, independent groups assumption
1.
Two
Sample TInterval:
We wish to find a confidence interval that is likely with the 95% confidence level to contain the true difference µ
G
 µ
B
between the mean lifetime of the generic brand
AA batteries and the mean lifetime of the brand name batteries.
In this situation, we use a computer to calculate the degrees of freedom because they are
approximately that of Student’s t. (Statistic +/ a Margin of Error)
= Independent Groups assumption, Randomization Condition, Nearly normal Condition
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 Spring '07
 VELLEMANP
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, Assumptions and Conditions

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