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Types of Control
I. Measurement Control
(Reliability and Validity)
II. Statistical Control
(External Validity)
III. Experimental Control
(Internal Validity)
Statistical Control - Sampling
I The Vocabulary of Sampling
II Types of Samples
III Testing Samples
IV Determining Sample Size
Statistical Control - Sampling
I. The Vocabulary of Sampling
A. The Universe
(The Theoretical Concept)
B. The Population
(The Indicator Concept)
C. The Sampling Frame (The Operational Procedure)
D. The Sample
(The Observational Set)
Statistical Control - Sampling
II. Types of Samples
A. Non-Probability
B. Probability
Statistical Control - Sampling
II. Types of Samples
A. Non-Probability
(Good for Exploratory Research)
1. Convenience Sampling
2. Referral Sampling
3. Quota Sampling
Statistical Control - Sampling
II. Types of Samples
A. Non-Probability
B. Probability (Good for Explanatory Research)
1. Simple Random Sample
2. Systematic Random Sample
3. Stratified Random Sample
4. Cluster or Area Sample
Statistical Control - Sampling
III. Testing a Single Sample Mean
A. The Central Limit Theorem - When random samples of
Size N are repeatedly taken from a population (no matter what
shape the population is, the resulting sampling distribution of
means is normal in shape with a mean equal to the population mean
and a standard deviation equal to the population S.D. divided by the
square root of N.
B. The Normal Curve
Statistical Control - Sampling
III. Testing Samples (cont.) - Hypothesis Testing:
A. State the Hypotheses
The Null Hypothesis (i.e. Ho)
The Alternative or Research Hypothesis (i.e. Ha)
B. Specify the Distribution
(e.g. Normal Distribution)
C. Set the Decision Criteria
Type I error – Alpha or Critical Region
D. Calculate the Outcome
(e.g. determine Z-value)
E. Make the Decision
Reject Ho or Do not Reject Ho
Statistical Control - Sampling
III. Testing Samples (cont.) - Single Sample Statistical Tests
The z-test is called for when both the
population mean and variance are known
and a point (mean value) is being evaluated
Example
The t-test is called for when the mean, but
not the population variance, is known and
a point (mean value) is being evaluated.
Example
Confidence Intervals are used when neither the
mean nor variance of the population are known
and an interval estimate is being evaluated.
Example
z
X 

N
X 
t
sˆ
N
 X z

N
Statistical Control - Sampling
IV. Sample Size (Three Components)
A. Error in Prediction
B. Confidence Level
C. Variability in Population
Click here to see formulae
Determining Sample Size
 X z
Consider the confidence interval
Solving for the sample size N we get
  X 
2
If
1
z
2

N
Then
N
  X    z

N

z  
N
2
2

2
  X 
2
Observe now the three parentheses that define N: 1) our confidence
level (z); 2) the population variance; and 3) the margin of error.
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Single Sample z-test
Problem: Suppose you want to test the idea that CSUN student GPAs are
higher this year than in the past. If records indicate that the past GPA is 2.50
with a S.D. of .5, can you conclude that a sample of 25 students, whose GPA
is 3.00, is significantly higher?
Solution: Use the five step hypothesis testing procedure
Step 1 State the hypotheses:
Ho:
 = 2.50
H1:  > 2.50
Step 2: Specify the distribution: Normal (Z-distribution)
Step 3: Set alpha (say .05; one tail test therefore Z= 1.65)
Step 4: Calculate the outcome:
Z
X 

N

3.0  2.5
.5
25
Step 5: Draw the conclusion: Reject Ho: 5.0 > 1.65
Students have a better GPA today.
= .5/.1 = 5.0
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Single Sample t-test
Problem: Given historical data, documenting the time it takes students at CSUN
to complete the undergraduate degree, yields an average of 6.5 years. Could you
conclude that students this semester take significantly longer to graduate if your
sample of 64 students this semester, yields a mean of 7.5 years, with a unbiased
standard deviation of 1.6 years?
Solution: Use the five step hypothesis testing procedure
Step 1 State the hypotheses:
Ho:

= 6.50 H1:

> 6.50
Step 2: Specify the distribution: Student’s t-distribution
Step 3: Set alpha (say .05; one tail test; therefore since N>30, t= 1.65)
Step 4: Calculate the outcome:
X 
t
sˆ
N

7.5  6.5
1.6
64
Step 5: Draw the conclusion: Reject Ho: 5.0 > 1.65
Students today take longer.
= 1.0/.2 = 5.0
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Confidential Intervals
Problem: Suppose you reject the null hypothesis in the z-test example and
now do not have a specific value of the population to use. Can we estimate
an interval within which we can be a certain degree assured that the actual
population falls?
Solution: Construct a confidence interval to estimate the population mean.
If the formula for the z-test is:
Z 
X 

N
Then solving for the population
mean we get the following:
Thus, in the previous example the
95% confidence interval would be:
 X z

N
  3.0  .196
3.0  1.96
 .5

25 

Thus, we can be 95% sure the true population
mean falls between the values 2.804 and 3.196
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