Chapter 8 Single Sample Tests Part II: Introduction to Hypothesis Testing Renee R. Ha, Ph.D. James...
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Transcript of Chapter 8 Single Sample Tests Part II: Introduction to Hypothesis Testing Renee R. Ha, Ph.D. James...
Chapter 8Single Sample Tests
Part II: Introduction to Hypothesis Testing
Renee R. Ha, Ph.D.James C. Ha, Ph.DIntegrative Statistics for the Social & Behavioral Sciences
Single Sample z-statistic
Mean Score Formula
z = X
X
X
Mean Score Formula Simplified
z = XX
Figure 8.1
When do we use the z-test?
1. When the experiment involves a single sample mean and the parameters of the corresponding null hypothesis population are known (μ and σ).
When do we use the z-test?
2. When the sampling distribution is normally distributed, which is the case if:
a. n ≥ 30 or
b. The null hypothesis population of raw scores is known to be normally distributed.
Single Sample t test
Comparison of Single Sample z and Single Sample t Formulas
z = XX =
n
X
t = XsX =
sn
X
Single Sample t testThe t critical values are dependent on sample size
(n) Now estimating from s, and the accuracy of σ
that estimation is dependent on n.Our critical probability values are now going to vary
with our sample size, unlike the z distribution in which the probabilities were independent of sample size.
Single Sample t testNew distribution called the t distribution.
It will vary with degrees of freedom (df), which are
related to sample size.
Single Sample t testDegrees of freedom: The number of scores that are
“free to vary” when calculating a statistic. The remaining value or values are then fixed.
Single Sample t testScore (X) Sample Mean ( ) Deviation
3 6 -3
5 6 -1
10 6 FIXED VALUE = 4
∑ Deviations = 0
X
18X
Figure 8.2Flowchart for choosing the appropriate test
Confidence IntervalsConfidence interval for the population mean: Range of
values with a calculated probability of containing the mean
Confidence IntervalsConfidence limits for the population mean: The upper
and lower values (or boundaries) surrounding the confidence interval.
Figure 8.3Graphic Display of a 95% Confidence Interval
Confidence IntervalsConfidence limits for the population mean: The upper
and lower values (or boundaries) surrounding the confidence interval.
μLower = X - s X (t 0.025 ) μUpper = X + s X (t 0.025 )
Effect Sizes and PowerEffect size is a standardized measure of the difference
between two (or more) group means; it is the difference in means divided by the shared standard deviation of two or more groups.
Effect Sizes and PowerFor a population with a known standard deviation ( ), σ
you can use Cohen’s d to calculate the effect size.
Then you can plug your effect size into a software package to calculate power.
Cohen’s D
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