Transcript 10.2
10.2: Estimating a Population
Mean (no )
• You are estimating the standard deviation,
but there will likely be some error
involved because you are estimating it
from sample data.
• Resulting distribution = t distribution
• Specify a particular t distribution by
degrees of freedom
• Notation: t(k)
T distribution features
• Similar in shape to the
standard Normal curve
• The spread is a bit greater
than the standard Normal
curve
• As degrees of freedom (k)
increases, the t(k) density
curve approaches the
N(0,1) curve more closely
Got Milk?
A milk processor monitors the number of bacteria per
milliliter in raw milk received for processing. A
random sample of 10 one-milliliter specimens from
milk supplied by one producer gives the following
data:
5370 4890 5100 4500 5260 5150 4900 4760
4700 4870
Find a 90% CI for the mean number of bacteria per
milliliter in all the milk from this supplier.
When degrees of freedom are
hard to find
• When the actual df does
not appear in Table C, use
the greatest df available
that is less than your
desired df.
• Rounding down in Table
C will widen the interval
which will safely contain
the exact interval.
• Recall:
Matched pairs are a form of block design in
which just two treatments are compared.
Subjects are matched in pairs and each
treatment is given to one subject in each pair
Alternatively, each subject receives both
treatments in some order.
• Comparative studies are more convincing than
single-sample investigations
Ex. 10.10, p. 651
• Our subjects are 11 people diagnosed as being
dependent on caffeine. Each subject was barred
from caffeine; instead, they took capsules
containing their normal caffeine intake. During a
different time period, they took placebo capsules.
The order in which subjects took caffeine and the
placebo was randomized.
• “Depression” is the score on the Beck Depression
Inventory. Higher scores = higher depression. We
are interested in whether or not caffeine
deprivation affects these outcomes.
• Construct a 90% confidence interval for the mean
change in depression score.
Subject
1
2
3
4
5
6
7
8
9
10
11
Depression Depression
(caffeine) (placebo)
5
5
4
3
8
5
0
0
2
11
1
16
23
5
7
14
24
6
3
15
12
0
• Matched pairs
problem (same
individual received
both treatments)
• Study looking to
provide evidence
that withholding
caffeine from
caffeine-dependent
individuals may
lead to depression
Random selection vs. Random
assignment
• Experiments are rarely done on randomly
selected subjects; the purpose is often to
compare 2 treatments rather than to
generalize to a larger population.
• Random selection = allows us to generalize
• Random assignment = allows us to
compare treatments
National Fuelsaver Corporation manufactures the
Platinum Gasaver, a device they claim “may
increase gas mileage by 22%.” Here are the
percent changes in gas mileage for 15 identical
vehicles, as presented in one of the company’s
advertisements:
48.3 46.9 46.8 44.6 40.2 38.5 34.6 33.7
28.7 28.7 24.8 10.8 10.4 6.9 12.4
1. Construct and interpret a 90% confidence
interval to estimate the mean fuel savings in the
population of all such vehicles. Follow the
Inference Toolbox.
2. Explain what “90% confidence” means in this
setting.
3. Comment on the manufacturer’s claim based on
your work in Question 1.