Mar. 31 Statistic for the day: Average number of baseball gloves that can be made from one cow: 5

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Mar. 31 Statistic for the dayAverage number of
baseball gloves that can be made from one cow 5

• Assignment
• Read Chapter 20 (again!)
• Do Exercises 4, 6, 8, 9, 10, 15

These slides were created by Tom Hettmansperger
and in some cases modified by David Hunter
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True or False?

• To construct a confidence interval for a
  population PROPORTION, it is enough to know the
  sample proportion and the sample size.
• To construct a confidence interval for a
  population MEAN, it is enough to know the sample
  mean and the sample size.

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Do each of the following tend to make a
confidence interval WIDER or NARROWER?

• A larger sample size
• A larger confidence coefficient
• A larger standard error of the mean
• A sample proportion closer to .5
• A larger sample mean

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• Thought questions
• In example 1 p352, for men who diet but do not
  exercise
• a 95 confidence interval for mean weight loss is
• 13 to 18 pounds.
• For men who exercise but do not diet the 95
  confidence
• interval for mean weight loss is
• 6 to 11 pounds.
• Do you think this means that 95 of all men who
  diet
• will lose between 13 and 18 pounds?
• On the basis of these results, do you think that
  you can
• conclude that men who diet without exercise lose
  more
• weight on average?

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Back to holding babies on the left. Accepting
that Lee Salk has presented a strong case for
holding babies on the left, what is the
selective advantage from the point of view of
evolution? Hypothesis Holding baby on the left
is holding baby over the heart. And the sound of
a human heartbeat is soothing to baby. To test
this hypothesis, Salk randomly selected a
period of 4 days and played the sound of a heart
beating in a new baby nursery. Then he did the
same without the heartbeat for a new group of
newborns.
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• Babies were divided into three groups
• light birth weight (2510 3000 grams),
• medium birth weight (3010 3500 grams) and
• heavy (3510 grams and above).
• The weight change from day 1 to day 4 was
  recorded.
• We want to know if the population means for
  heartbeat
• and control (no heartbeat) are close or not.

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We dont know the population means. We take
samples and compute the sample means. Since the
sample means are attracted to the population
means, we want to check to see if the sample
means are close. How do we decide if the sample
means are close? One way compare the 95
confidence intervals. If the two confidence
intervals are separated then perhaps we can
conclude that the population means are
separated and are not close. Using confidence
intervals takes uncertainty due to variation in
the sample means into account.
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We informally compare the 95 confidence
intervals to try to decide if the population
means are close or not. In this case we conclude
that for all three birth weights the population
mean weight change for heartbeat babies is
greater than the population mean weight change
for the control babies. Is there a more formal
way to approach the question Is there a
difference between the population means?
Construct a 95 confidence interval for the
difference in population means based on the
difference in sample means.
YUP!
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First, a digression.
Suppose I tell you that I have given IQ tests to
a sample of PSU students. I tell you the mean
IQ for the sample is 105. Question Is this
close to 100 or not?
What other information do you need in order to
answer the question?
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Suppose you ask me for the SEM.
It is 2. Now is 105 close to 100 or not?
Why?
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Suppose you ask me for the SEM.
It is 4. Now is 105 close to 100 or not?
Why?
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• So to answer the question Is a sample mean
• of 105 close to 100 or not, you need the
• SEM (standard error of the mean).
• I could give it to you directly. OR
• I could give you
• the sample size and
• the sample standard deviation, SD
• Then SEM SD/sqrt(sample size)
• For example sample size 100 and SD 20
• Then SEM 20/10 2.

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Now think of TWO sample means
Suppose I have two sample means and I want to
know if they are close to each other. This is
equivalent to Is the difference between the two
sample means close to zero? Let D denote the
difference in sample means. What do you need
from me to decide if the difference D is close to
zero?
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• You need the standard deviation of the difference
  of sample means.
• Example
• Suppose I tell you that I have two samples of
  babies, one that listened to heartbeats and the
  other that did not.
• I measure weight gained and tell you
• Heartbeat group sample mean weight gain is 65 g
• Control group sample mean weight gain is -20 g
• Are the sample means close? Is the difference of
  85 grams close to 0?

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I now tell you that the standard deviation of the
difference in sample means is 14.13 g Can you
tell if the difference of 85 g is close to 0?
You need to check to see if 0 is within 2
standard deviations of 85 (we suppress the normal
curve) 85 2x(14.13) 85 28.26 56.74 to
113.26 So 0 is not close to 85 in this case and
we conclude that the sample means are not close
to each other.
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Recall the Pythagorean Theorem
C sqrt( A2 B2)
A
B
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Question How can we get the standard deviation
of the difference from information on the two
samples?

• Suppose we have the SEMs for the two sample
  means
• Heartbeat SEM 8.45 g
• Control SEM 11.33 g

Sqrt( 8.452 11.332) 14.13
Heartbeat SEM 8.45
Control SEM 11.33
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To decide if two sample means are
close, we check to see if their
difference is close to 0. We must have
the standard deviation of the difference.
Once we have that we can check to see if 0 is
within 2 standard deviations of the
difference. We could be given The individual
sample mean SEMs Then compute the standard
deviation of the difference using the Pythagorean
theorem. The individual sample sizes and the
individual sample standard deviations. Then
compute the individual SEMs from SD/sqrt(sample
size) Then compute the standard deviation of the
difference using the Pythagorean theorem.