# T-test: Arithmetic Mean and Statistical Power

Matched T-test (four steps of NHT)
Step 1
Scientific (just the written portion)
Null (no effect)
Alternative (there is an effect)
Step 2
Rather than finding standard error we look for standard deviation first. Standard error has a parallel formula to SD. Smd = SD/ sqrt n. Step 3
Set Alpha to .05 two tailed
Critical value of t. (t table)
Decision rules, either sketch or write out.
Step 4
Conclusion
Mean of difference scores
T-statistic takes the same basic form (statistic minus expected value/SD) Reported as t(9) = .85, n.s.
Statistical decision (don’t reject null all hypothesis are plausible; reject accept all alternative hypotheses) Interpretation

Independent group t-tests
The logic of testing hypothesis about the means of two independent groups is the same as for previous statistical tests Some minor calculation differences that can seem difficult at first The test provides a more detailed discussion of the standard deviation The equation for any test may be thought of as three parts

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Sample statistic
Expected value (if H0 is true)
A measure of the variability in the sample statistic
H0 is written as the difference between two means
Two assumptions greatly simplify equations
Homogeneity of Variance: it is assumed that variance in population 1 Is equal to the variance in population 2. IMPORTANT!!!
The assumption regards the population variances, not sample variances. It is possible that s21 is not equal to s22
Second assumption… Normality

CI for a single mean
For a one sample t-test
CI = M +/- (t-critical) (sm)
 Critical value was a function of df and desired level of confidence
The logic of a CI for the difference between two means is identical to single group mean We are 95% confident the population means of the difference scores between husbands and wives in the population lies within the range of -1.669 to 3.669.

CI for Independent-Groups t-test
CI[0.84 ≤ μR-μC ≤ 9.26] = .95. Based on these data, we can say we are 95% confident that the mean difference between reward and no reward conditions in the population lies within the range of 0.84 to 9.16. Note this could be thought of as the “effect” of treatment.

Statistical Power
Statistical power (and type II error rate) discussed with respect to hypothesis testing
Ideally all studies should have high power
It has been recommended that power should be .8 or greater

Five feature that’s increase power
1.Increasing alpha
2.Using one tailed, rather than two-tailed, test
3.Increasing magnitude of the effect (“size of treatment effect”) 4.Decreasing variability in the outcome
5.Increasing sample size
First two not seen as viable methods to increase power.

Consideration for Power in figures
Considering sampling distributions can help illustrate the interplay between statistical power and the five features defined above.

Distributions will be sampling distribution
They could be sampling distributions of the mean…

Estimating Statistical power
 Suggested that researchers should attempt to conduct experiments with power levels equaling approximately .80.

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