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One Sample t Tests

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One Sample t Tests. Karl L. Wuensch Department of Psychology East Carolina University. Nondirectional Test. Null:  = some value Alternative:   that value We have a sample of N scores Somehow we magically know the value of the population  - PowerPoint PPT Presentation
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One Sample t Tests Karl L. Wuensch Department of Psychology East Carolina University
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Page 1: One Sample  t  Tests

One Sample t TestsKarl L. Wuensch

Department of PsychologyEast Carolina University

Page 2: One Sample  t  Tests

Nondirectional Test• Null: = some value• Alternative: that value• We have a sample of N scores• Somehow we magically know the value of

the population • We trust that the population is normally

distributed• Or invoke the Central Limit Theorem

Page 3: One Sample  t  Tests

H0: IQ = 100 N = 25, M = 107, = 15

325

15 NM

p = .0198, two-tailed

33.23

100107

M

MZ

Page 4: One Sample  t  Tests

Directional Test• For z = 2.33• If predicted direction in H1 is correct, then

p = .0099• If predicted direction in H1 is not correct,

then p = 1 - .0099 = .9901

Page 5: One Sample  t  Tests

Confidence Interval

MM CVMCVMCI

88.11212.101)515(96.1107)515(96.1107

95.

CI

Page 6: One Sample  t  Tests

The Fly in the Ointment

• How could we know the value of but not know the value of ?

Page 7: One Sample  t  Tests

Student’s t

• The sampling distribution of 2 is unbiased but positively skewed.

• Thus, more often than not, s2 < 2 • And | t | > | z |, giving t fat tails (high

kurtosis)

NsMt

NMZ

Page 8: One Sample  t  Tests

Fat-Tailed t• Because of those fat tails, one will need go

out further from the mean to get to the rejection region.

• How much further depends on the df, which are N-1.

• The fewer the df, the further out the critical values.

• As df increase, t approaches the normal distribution.

Page 9: One Sample  t  Tests

CV for t, = .05, 2-tailedDegrees of Freedom Critical Value for t

1 12.7062 4.3033 3.182

10 2.22830 2.042

100 1.984 1.960

Page 10: One Sample  t  Tests

William Gosset

Page 11: One Sample  t  Tests

SAT-Math• For the entire nation, between 2000 and

2004, = 516.• For my students in undergrad stats:

M = 534.78s = 93.385N = 114

• H0: For the population from which my students came, = 516.

Page 12: One Sample  t  Tests

We Reject That Null

df = N – 1 = 113 p = .034

746.8114385.93

NssM

147.2746.8

51678.5340

Ms

Mt

Page 13: One Sample  t  Tests

CI.95

• From the t table for df = 100, CV = 1.984.

MM sCVMsCVMCI

13.55243.517)746.8(984.178.534)746.8(984.178.534

95.

CI

Page 14: One Sample  t  Tests

Effect Size

• Estimate by how much the null is wrong.• Point estimate = M – null value• Can construct a CI.• For our data, take the CI for M and

subtract from each side the null value• [517.43 – 516, 552.13 – 516] = • [1.43, 36.13]

Page 15: One Sample  t  Tests

Standardized Effect Size• When the unit of measure is not

intrinsically meaningful,• As is often case with variables studied by

psychologists,• Best to estimate the effect size in standard

deviation units.• The parameter is

Page 16: One Sample  t  Tests

Estimated

• We should report a CI for • Constructing it by hand in unreasonably

difficult.• Professor Karl will show how to use SAS

or SPSS to get the CI.

20.385.9378.18

s

Md

Page 17: One Sample  t  Tests

Assumptions• Only one here, that the population is

normally distributed.• If that is questionable, one might use

nonlinear transformations, especially if the problem is skewness.

• Or, use analyses that make no normality assumption (nonparametrics and resampling statistics).

Page 18: One Sample  t  Tests

Summary Statements• who or what the research units were

(sometimes called “subjects” or “participants”)

• what the null hypothesis was (implied)• descriptive statistics such as means and

standard deviations• whether or not you rejected the null

hypothesis

Page 19: One Sample  t  Tests

Summary Statements 2• if you did reject the null hypothesis, what

was the observed direction of the difference between the obtained results and those expected under the null hypothesis

• what test statistic (such as t) was employed

• the degrees of freedom

Page 20: One Sample  t  Tests

Summary Statements 3• if not obtainable from the degrees of

freedom, the sample size• the computed value of the test statistic• the p value (use SPSS or SAS to get an

exact p value)• an effect size estimate• and a confidence interval for the effect

size parameter

Page 21: One Sample  t  Tests

Example Summary Statements

• Carefully study my examples in my document One Mean Inference.

• Pay special attention to when and when not to indicate a direction of effect.

• and also when the CI would more appropriately be with confidence coefficient (1 - 2) rather than (1 - ).

Page 22: One Sample  t  Tests

The t Family


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