One-Sample and Two-Independent-Sample t-Tests: Lecture Notes, Study notes of Statistics for Psychologists

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Basic Statistics for

The Behavioral Sciences

LECTURE NOTES

Ch. 9. One-sample t-test I. Situation A. One sample (group); same as the z-test. B. We want to know if the μ of a population from which our sample was drawn is equal to a given μ-value (same as the z-test). C. The μ is given, but the σ is unknown. D. If we do not know σ and should use s, we should use t as a test statistic instead of z, and should use the t-dist as a reference dist instead of the z-dist.

E. M - μ t = ─────

s/ n F. Notice that the only difference from z is the s in the denominator. Everything else is exactly the same as z. 15

II. t-distribution; the t-dist is almost identical to the standard normal dist (SND) except the following. A. It has more variability than SND. B. It has one parameter, degrees of freedom. C. The shape of the dist depends on the df.

III. Degrees of freedom (df). A. Whenever we estimate an unknown fixed value, we have the freedom to choose n-1 values, but NOT the nth value. B. The number of components which we are free to choose in a sample when we estimate some unknown fixed parameter(s) is called degrees of freedom (df). C. df= # of independent components minus (-)

of parameters estimated.

IV. t-table A. Unlike the z-table, the numbers in t-table are critical values, not probabilities. B. Probabilities corresponding to critical values appear on the top of the table. C. We can use this table for the critical value approach decision rule.

D. If │tobs│≥tcrit, reject H 0 , otherwise, fail to reject H 0.

V. Procedure

IV. Example

V. Robustness (against the violation of assumptions) A. The strength of test statistics. B. The extend to which the sampling dist of emiprical data fits the theoretical dist.

C. Is αtrue ≈ αset? D. Is a test still good even if we violate the assumption(s)?

E. Summary table ┌───────────┬────────────────┬────────────────┐

│Assumption │ Do we typically│ If we don't, │ │ │ meet the │ is the test │

│ │ assumption? │ robust? │ ├───────────┼────────────────┼────────────────┤

│normality │ No │ Yes │

│σ1² = σ2² │ No │ Yes, if n1=n2 │

│ │ │ No, if n1≠n2 │

│indepen- │ Yes, if no, │ No │ │dence │ don't use t │ │

└───────────┴────────────────┴────────────────┘ For normality and equal variance, we can not observe them because they are in the population. For independence we can observe it in the sample.

VI. Strength of relationship

r²= t df

t

2

2 d = 2

1 2 s P

M − M