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Figure 7.1 (p. 163) - A population of scores
μ=5
Table 7.1 (p. 164) All possible random samples of n=2
Figure 7.2 The distribution of sample means is
centered around the population mean
The central limit theorem: For any population with a
mean (μ) and standard deviation (σ) the distribution of
sample means for sample size n will have a mean of μ
and a standard deviation of σ/√ n and will approach a
normal distribution as n approaches infinity.
16
14
Pop
Sample Mean (n=2)
Frequency
12
10
Sample Mean (n=4)
Sample Mean (n=8)
8
6
4
2
0
1
2
3
4
5
6
7
8
9
Hours of Sleep
What is happening to s as n gets larger?
10
11
12
• The standard deviation of the distribution of
sample means is called the standard error of
M (σM). This approaches zero as sample size
increases.
Pop.
N = 80
Samples
2
4
8
μ = 6.23
6.73
6.23
6.22
σ = 2.33
1.71
1.02
0.74
1.65
1.17
0.82
CL Theorem Predicts =
The values predicted by the central limit theorem are the
values that would have been attained if you had taken
every possible sample of n=2, 4, and 8. Ours were still
close.
• A population has a size of N= 4.
• What would be the approximate standard
deviation of sample means for eighty
samples of size (n=4)?
• Law of large Numbers: As sample size
increases, the sample means (M) will get
closer to the population mean.
• A population has a size of N= 4.
• What would be the approximate standard
deviation of sample means for eighty
samples of size (n=1)?
• Standard error (σM) will always be between
0 and the population standard deviation.
– It will be close to zero when sample sizes are
very large and equal to the population
standard deviation (σ) when sample sizes are
very small (n = 1).
σM = σ/√ n
• The distribution of sample means is used
to tell us about the probability associates
with a specific sample.
– If a sample is drawn from a known population,
how likely is it to have a particular mean?
Figure 7.5 (p. 171) The population of scores on the SAT has a
μ=500 and σ=100. What is the probability of drawing a
sample of n=25 with a M > 540?
What is σM?
Formula for sample mean z should look familiar
z = (M-μ)/(σM)
•
This is what it’s all about!! (almost)
1. You have developed the ultimate SAT prep
course!
2. Select 25 High School students at random
3. Have them take your course
4. Have them take the SAT
5. Determine their mean scores and see if they
are unusual.
6. Scientists generally consider unusual to be
associated with p < 0.05
– This is the core of hypothesis testing
• What kind of SAT scores can you expect
from a random sample (n=25) 80% of the
time?
σM = 20
Μ = 500
Figure 7.6 (p. 173)
Figure 7.7 (p. 174) Sampling error: Sample means will not
perfectly match the population means, but most will be close
Figure 7.8 (p. 175) Standard error: Gets smaller as
sample sizes get larger
Table 7.2 (p. 177) Reporting standard error with sample means
Figure 7.9 (p. 177)
Figure 7.10 (p. 177)
Figure 7.11 (p. 179)
Figure 7.12 (p. 180) A sample of 25 treated rats has a 95% chance
of being between 392.16 and 407.84