Where We Left Off

1
Where We Left Off

• What is the probability of randomly selecting a
  sample of three individuals, all of whom have an
  I.Q. of 135 or more?
• Find the z-score of 135, compute the tail region
  and raise it to the 3rd power.

So while the odds chance selection of a single
person this far above the mean is not all that
unlikely, the odds of a sample this far above the
mean are astronomical
z 2.19
P 0.0143
0.01433 0.0000029
X
X
2
Sampling Distributions

• I What is a Sampling Distribution?
• A If all possible samples were drawn from
  a population
• B A distribution described with Central
  Tendency µM
• And dispersion sM ,the standard error
• II The Central Limit Theorem

3
Sampling Distributions

• What youve done so far is to determine the
  position of a given single score, x, compared to
  all other possible x scores

x
4
Sampling Distributions

• The task now is to find the position of a group
  score, M, relative to all other possible sample
  means that could be drawn

M
5
Sampling Distributions

• The reason for this is to find the probability of
  a random sample having the properties you observe.

M
6
Sampling Distributions

• Any time you draw a sample from a population, the
  mean of the sample, M , it estimates the
  population mean µ, with an average error of
• We are interested in understanding the
  probability of drawing certain samples and we do
  this with our knowledge of the normal
  distribution applied to the distribution of
  samples, or Sampling Distribution
• We will consider a normal distribution that
  consists of all possible samples of size n from a
  given population

7
Sampling Error
Sampling error is the error resulting from using
a sample to estimate a population characteristic.
8
Sampling Distribution of the Mean
For a variable x and a given sample size, the
distribution of the variable M (i.e., of all
possible sample means) is called the sampling
distribution of the mean. The sampling
distribution is purely theoretical derived by the
laws of probability. A given score x is part of a
distribution for that variable which can be used
to assess probability A given mean M is part of a
sampling distribution for that variable which can
be used to determine the probability of a given
sample being drawn
9
The Basic Concept

• Extreme events are unlikely -- single events
• For samples, the likelihood of randomly selecting
  an extreme sample is more unlikely
• The larger the sample size, the more unlikely it
  is to draw an extreme sample

10
The original distribution of x 2, 4, 6, 8
Now consider all possible samples of size
n 2What is the distribution of sample means M
11
The Sampling Distribution For n2Notice that
its a normal distribution with µ 5
12
Heights of the five starting players
13
Possible samples and sample means for samples of
size two
M
14
Dotplot for the sampling distribution of the mean
for samples of size two (n 2)
M
15
Possible samples and sample means for samples of
size four
M
16
Dotplot for the sampling distribution of the mean
for samples of size four (n 4)
M
17
Sample size and sampling error illustrations for
the heights of the basketball players
18
Dotplots for the sampling distributions of the
mean for samples of sizes one, two, three, four,
and five
M
M
M
M
M
19
Sample Size and Standard Error
The possible sample means cluster closer around
the population mean as the sample size increases.
Thus the larger the sample size, the smaller the
sampling error tends to be in estimating a
population mean, m, by a sample mean, M. For
sampling distributions, the dispersion is called
Standard Error. It works much like standard
deviation.
20
Standard Error of M
For samples of size n, the standard error of the
variable x equals the standard deviation of x
divided by the square root of the sample
size In other words, for each sample size,
the standard error of all possible sample means
equals the population standard deviation divided
by the square root of the sample size.
21
The Effect of Sample Size on Standard ErrorThe
distribution of sample means for random samples
of size (a) n 1, (b) n 4, and (c) n 100
obtained from a normal population with µ 80 and
s 20. Notice that the size of the standard
error decreases as the sample size increases.
22
Mean of the Variable M
For samples of size n, the mean of the variable M
equals the mean of the variable under
consideration mM m. In other words, for each
sample size, the mean of all possible sample
means equals the population mean.
23
The standard error of M for sample sizes one,
two, three, four, and five
Standard error dispersion of MsM
24
The sample means for 1000 samples of four IQs.
The normal curve for x is superimposed
25
Sampling Distribution of the Mean for a Normally
Distributed Variable
Suppose a variable x of a population is normally
distributed with mean m and standard deviation s.
Then, for samples of size n, the sampling
distribution of M is also normally distributed
and has mean mM m and standard
error of
26
(a) Normal distribution for IQs(b) Sampling
distribution of the mean for n 4(c) Sampling
distribution of the mean for n 16
27
Samples Versus Individual Scores
28
Frequency distribution for U.S. household size
29
Relative-frequency histogram for household size
30
Sample means n 3,for 1000 samples of household
sizes.
31
The Central Limit Theorem
For a relatively large sample size, the variable
M is approximately normally distributed,
regardlessof the distribution of the underlying
variable x. The approximation becomes better and
better with increasing sample size.
32
Sampling distributions fornormal, J-shaped,
uniform variable
M
M
M
M
33
APA Style TablesThe mean self-consciousness
scores for participants who were working in front
of a video camera and those who were not
(controls).
34
APA Style Bar GraphsThe mean (SE) score for
treatment groups A and B.
35
APA Style Line GraphsThe mean (SE) number of
mistakes made for groups A and B on each trial.
36
Summary

• We already knew how to determine the position of
  an individual score in a normal distribution
• Now we know how to determine the position of a
  sample of scores within the sampling distribution
• By the Central Limit Theorem, all sampling
  distributions are normal with

37
Sample Problem 1

• Given a distribution with µ 32 and s 12 what
  is the probability of drawing a sample of size 36
  where M gt 48

Does it seem likely that M is just a chance
difference?
38
Sample Problem 2

• In a distribution with µ 45 and s 45 what is
  the probability of drawing a sample of 25 with M
  gt50?

39
Sample problem 3

• In a distribution with µ 90 and s 18, for a
  sample of n 36, what sample mean M would
  constitute the boundary of the most extreme 5 of
  scores?
• zcrit 1.96

40
Sample Problem 4

• In a distribution with µ 90 and s 18, what is
  the probability of drawing a sample whose mean M
  gt 93?

What information are we missing?
n 9
41
Sample Problem 5

• In a distribution with µ 90 and s 18, what is
  the probability of drawing a sample whose mean M
  gt 93?

n 16
42
Sample Problem 6

• In a distribution with µ 90 and s 18, what is
  the probability of drawing a sample whose mean M
  gt 93?

n 25
43
Sample Problem 7

• In a distribution with µ 90 and s 18, what is
  the probability of drawing a sample whose mean M
  gt 93?

n 36
44
Sample Problem 8

• In a distribution with µ 90 and s 18, what is
  the probability of drawing a sample whose mean M
  gt 93?

n 81
45
Sample Problem 9

• In a distribution with µ 90 and s 18, what is
  the probability of drawing a sample whose mean M
  gt 93?

n 169
46
Sample Problem 10

• In a distribution with µ 90 and s 18, what is
  the probability of drawing a sample whose mean M
  gt 93?

n 625
47
Sample Problem 10

• In a distribution with µ 90 and s 18, what is
  the probability of drawing a sample whose mean M
  gt 93?

n 1
48
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Sample Problem 11
z 2.58

• In a distribution with µ 200 and s 20, what
  sample mean M corresponds to the most extreme 1 ?

n 1
50
Sample Problem 12
z 2.58

• In a distribution with µ 200 and s 20, what
  sample mean M corresponds to the most extreme 1 ?

n 4
51
Sample Problem 13
z 2.58

• In a distribution with µ 200 and s 20, what
  sample mean M corresponds to the most extreme 1 ?

n 16
52
Sample Problem 14
z 2.58

• In a distribution with µ 200 and s 20, what
  sample mean M corresponds to the most extreme 1 ?

n 64
53
Sample Problem 15
z 2.58

• In a distribution with µ 200 and s 20, what
  sample mean M corresponds to the most extreme 1 ?

n 258
54
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• n 1

150 175 200 225
250
z -2.58
2.58 M 148 .4

251.6
56

• n 4

150 175 200 225
250
z -2.58 2.58 M
174.2 225.8
57

• n 16

150 175 200 225
250
z -2.58 2.58 M 187.1 212.9
58

• n 64

150 175 200 225
250
z -2.58 2.58 M 193.6 206.4
59

• n 258

150 175 200 225
250
z -2.58 2.58 M 196.8 203.2
60

• n 8

150 175 200 225
250