Chapter 6
Samples and Populations
Samples and Populations
• Population:
– Set of individuals who share at least one
characteristic
• Sample:
– A smaller number of individuals from the
population that share one or more given
characteristics
Sampling Methods
• Researchers want to make inferences from
data.
• Sampling methods vary according to
population and access.
• Two types of sampling methods:
– Non-random samples
– Random samples
Sampling Error
• Mean of a sample shown as X
• Mean of a population shown as µ
• Standard deviation of a sample shown as s
• Standard deviation of a population shown as σ
• Mean or standard deviation of a sample rarely
identical to the population
– This difference is known as sampling error.
Table 1: A population and three random
samples of final exam grades
Population
70
80
93
86
85
90
56
52
67
40
78
89
Sample A
Sample B
Sample C
96
40
72
57
99
86
96
49
48
56
56
49
99
72
30
52
67
56
96
94
μ = 71.55
N = 20
X = 75.75 X = 62.25 X = 68.25
Sampling Distributions of Means
96
93
92
98
96
100
106
102
105
99
103
107
101
102
104
91
108
95
7
Figure 1: The Mean Long Distance Phone Time in 100 Random Samples in
Standard Error of the Mean
• Standard deviation of theoretical sampling
distribution can be derived.
• This is known as the standard error of the mean.
• Formula for standard error of the mean:

X 
N
• We can now calculate the range of mean values in
which our population mean is likely to fall.
Standard Error of Mean
• Obtained by dividing the population
standard deviation by the square root of
the sample size
– Illustration: IQ test
•
•
Population mean of 100
Population standard deviation of 15
– If we took a sample of 10, subject to a standard error
of?
• With the aid of the standard error of the
mean, we can find the range of mean
values within which our true population
mean is likely to fall.
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Confidence Interval Cont.
•
•
Can be constructed for any level of probability
It is has become a matter of convention to use a
wider, less precise confidence interval having a
better probability of making an accurate or true
estimate of the population mean.
– 68% confidence interval = X ± (1) X
– 95% confidence interval = X ± (1.96)  X
– 99% confidence interval = X ± (2.58)  X
Confidence Interval Cont.
•
•
How do we go about finding the 95% confidence
interval?
We already know that roughly 95% of the
sample means in a sampling distribution lie
between -2 standard deviations and +2 standard
deviations from the mean of means.
11
95% Confidence Interval Using z
• Suppose we want to determine the expected
miles per gallon for a new Ford Explorer?
– Standard deviation = 4 miles/gallon
– N = 100 cars
– Sample Mean = 26 miles/gallon
• How do we obtain a 95% confidence interval for
the mean miles/gallon for all cars of this model?
– What would happen if we only used 20 cars for our
sample?
12
99% Confidence Interval Using z
• Now, the statistician is informed that 95%
confidence is not confident enough for
their needs. To be confident, we want 99%.
– Standard deviation = 4 miles/gallon
– N = 20 cars
– Sample Mean = 26 miles/gallon
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End Day 1
The t ratio
•
•
•
•
•
Very few situations in which the population standard deviation (and
thus the standard error of the mean) is known
When the exact standard deviation of the population (σ) is
unknown, the t-distribution is used
Recall that sample means (and their standard deviations) are lower
and more stable than population means
It is then necessary to inflate the sample standard deviation to
produce more accurate estimates
Standard Error for a t ratio
sX 
s
N 1
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Degrees of Freedom
• The greater the degrees of freedom, the larger the
sample size and the closer the t distribution gets to
the normal distribution
• df = N – 1
• Recall that the only difference between a t and a z is
that the former uses an estimate of the standard
error based on sample data while the latter is known
• What would one do for larger samples for which the
degrees of freedom may not appear in Table C?
16
The t ratio
• For t-distributions, use Table C instead of Table A
– Various levels of alpha
• Alpha = area in the tails of the t distribution
– For a 95% level of confidence, an alpha = .05.
– For a 99% level of confidence, an alpha = .01.
• With the addition of alpha, we now have two pieces of
information available and can now construct our
confidence interval
– Degrees of freedom (N – 1)
– Alpha value (95% = .05 or 99% = .01)
𝐶𝐼 = 𝑥 ± (𝑡)(𝑠𝑥 )
17
Putting it all together
A local newspaper surveyed 20 citizens on its coverage of the
recent rape case in Stuebenville, Ohio. In regards to how fair
the coverage was to the victim, the newspaper used a scale
ranging from 1 (completely fair) to 10 (completely unfair).
Construct a 95% and 99% confidence interval with a sample of
20, a mean of 7.2, and a standard deviation of 1.7.
Suppose that a researcher wanted to examine the extent of
cooperation among kindergarten children. To do so, she
unobtrusively observes 9 children at play for 30 minutes and
notes the number of cooperative acts engaged in by each child:
The mean number of cooperative acts was 2.67 and the
standard deviation was 1.32.
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Estimating Proportions
• We can also estimate population proportions.
• Pattern of formulas is the same.
sP 
P(1  P)
N
•sP = standard error of the proportion
•P = sample proportion
•N = total number in the sample
We either use:
95% CI = P ± 1.96*sp
or
99% CI = P ± 2.58*sp
An Illustration
• Suppose a polling organization contacted 400 members of a local
police union and asked them whether they intended to vote for
candidate A or candidate B. Suppose that 55% reported their
intention to vote for candidate A. Find the 95% confidence interval
for candidate A.
Step 1: Obtain the standard error of the proportion.
Step 2: Multiply the standard error of the proportion by 1.96 to obtain
the margin of error.
Step 3: Add and subtract the margin of error to find the confidence
interval.