Using t-tests Basic introduction and 1-sample t-tests Statistics for the Social Sciences Psychology...

Using t-tests Basic introduction and 1-sample t-tests

Statistics for the Social SciencesPsychology 340

Spring 2010

PSY 340Statistics for the

Social SciencesOutline (for content up to Exam 2)

• Review t-tests (note: next exam is March 4, right before spring break)

– One sample, related samples, independent samples

– Effect sizes with t-tests

– Confidence intervals in t-test type designs


Social Sciences

Statistical analysis follows design

• The one-sample z-test can be used when:– 1 sample

€

zX

=X − μ

X

σX

– One score per subject

– Population mean (μ) and standard deviation (σ)are known


Social Sciences

Statistical analysis follows design

• The one-sample t-test can be used when:– 1 sample

– One score per subject

– Population mean (μ) is known

– but Population standard deviation (σ) is NOT known

€

t =X − μ

X

sX


Social Sciences Testing Hypotheses

– Step 1: State your hypotheses

– Step 2: Set your decision criteria

– Step 3: Collect your data

– Step 4: Compute your test statistics • Compute your estimated standard error

• Compute your t-statistic

• Compute your degrees of freedom

– Step 5: Make a decision about your null hypothesis

• Hypothesis testing: a five step program


Social Sciences Performing your statistical test

• What are we doing when we test the hypotheses?– Consider a variation of our memory experiment example

Population of memory patients

MemoryTestμ is known

Memory treatment

Memory patients

MemoryTest

X

Compare these two meansConclusions:

• the memory treatment sample are the same as those in the population of memory patients.• they aren’t the same as those in the population of memory patients

H0:

HA:



• What are we doing when we test the hypotheses?

Real world (‘truth’)

H0: is false (is a treatment effect)

Two populations

XA

they aren’t the same as those in the population of memory patients

H0: is true (no treatment effect)

One population

XA

the memory treatment sample are the same as those in the population of memory patients.



• What are we doing when we test the hypotheses?– Computing a test statistic: Generic test

€

test statistic =observed difference

difference expected by chance

Could be difference between a sample and a population, or between different samples

Based on standard error or an estimate of the standard error



€



€

t =X − μ

X

sX

€

zX

=X − μ

X

σX

Test statistic

One sample z One sample tidentical



€



€

t =X − μ

X

sX

€

zX

=X − μ

X

σX

Test statistic

Diff. Expected by chance

€

σX

=σ

nStandard error

One sample z One sample t

different

don’t know this,so need to estimate it



€



€

t =X − μ

X

sX

€

zX

=X − μ

X

σX

€

sX

=s

n

Test statistic


€

σX

=σ

nStandard error

Estimated standard error


different

“Unbiased estimate of the population standard

deviation”This is the same as the

“sample standard deviation”



€



€

t =X − μ

X

sX

€

zX

=X − μ

X

σX

€

sX

=s

n

Test statistic


€

σX

=σ

nStandard error

Estimated standard error


different

don’t know this,so need to estimate it

€

df = n −1Degrees of freedom


Social Sciences

Proportion in one tail0.10 0.05 0.025 0.01 0.005

Proportion in two tailsdf 0.20 0.10 0.05 0.02 0.011 3.078 6.314 12.706 31.821 63.6572 1.886 2.920 4.303 6.965 9.9253 1.638 2.353 3.182 4.541 5.8414 1.533 2.132 2.776 3.747 4.6045 1,476 2.015 2.571 3.365 4.0326 1.440 1.943 2.447 3.143 3.707:

15:

:1.341

:

:1.753

:

:2.131

:

:2.602

:

:2.947

:

One sample t-test

• The t-statistic distribution (a transformation of the distribution of sample means transformed)– Varies in shape according to the degrees of freedom

• New table: the t-table


Social Sciences



15:

:1.341

:

:1.753

:

:2.131

:

:2.602

:

:2.947

:

One sample t-test

If test statistic is here Fail to reject H0

Distribution of the t-statistic

If test statistic is here Reject H0

– To reject the H0, you want a computed test statistics that is large

• The alpha level gives us the decision criterion

• New table: the t-table

• The t-statistic distribution (a transformation of the distribution of sample means transformed)


Social Sciences



15:

:1.341

:

:1.753

:

:2.131

:

:2.602

:

:2.947

:

One sample t-test

• New table: the t-tableOne tailed

- or - Two-tailed

Critical values of ttcrit

Degrees of freedomdf

α - level


Social Sciences



15:

:1.341

:

:1.753

:

:2.131

:

:2.602

:

:2.947

:

One sample t-test

• What is the tcrit for a two-tailed hypothesis test with a sample size of n = 6 and an α-level of 0.05?

Distribution of the t-statisticα = 0.05Two-tailed

n = 6

df = n - 1 = 5

tcrit = +2.571


Social Sciences



15:

:1.341

:

:1.753

:

:2.131

:

:2.602

:

:2.947

:

One sample t-test

Distribution of the t-statisticα = 0.05One-tailed

n = 6

df = n - 1 = 5

tcrit = +2.015

• What is the tcrit for a one-tailed hypothesis test with a sample size of n = 6 and an α-level of 0.05?


Social Sciences One sample t-test

An example: One sample t-test

Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment.

• How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60?

• After the treatment they have an average score of = 55, s = 8 memory errors.

€

X

Don’t know σ

Do know s







€

X

• Step 1: State your hypotheses

H0: the memory treatment sample are the same (or worse) as those in the population of memory patients.

HA: they perform better than those in the population of memory patients

μTreatment > μpop = 60

μTreatment < μpop = 60



Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment. • Step 2: Set your decision

criteria

H0: μTreatment > μpop = 60 HA: μTreatment < μpop = 60

α = 0.05One -tailed



€

X





Memory experiment example:• We give a n = 16 memory patients a memory improvement treatment. • Step 2: Set your decision

criteriaα = 0.05One -tailed



€

X







• Step 3: Collect your data



€

X







• Step 4: Compute your test statistics

€

t =X − μ

X

sX

€

=55 − 60

816

⎛ ⎝ ⎜

⎞ ⎠ ⎟

= -2.5



€

X


sX =sn=

816






• Step 4: Compute your test statistics

€

df = n −1

t = -2.5



€

X

€

=16 −1=15







€

t(15) = −2.5

• Step 5: Make a decision about your null hypothesis



€

X

€

df =15


Proportion in two tailsdf 0.20 0.10 0.05 0.02 0.01:

15:

:1.341

:

:1.753

:

:2.131

:

:2.602

:

:2.947

:

tcrit = -1.753







€

t(15) = −2.5

• Step 5: Make a decision about your null hypothesis



€

X

€

df =15

-1.753 = tcrit

tobs=-2.5

- Reject H0



Social Sciences Assumptions of t-statistics

• Values in the sample must be independent observations – Each observation is independent of all of the other

observations (see pg 253 of your textbook for examples of non-independence)

• The population that is sampled must be normally distributed– It turns out that t-tests are pretty robust against

violations of this assumption, especially with large samples


Social SciencesEffect Sizes & Power for 1-sample t Test

d =μ1 −μ2

σ

Recall for the 1-sample z-test:

Remember we don’t know these with the t-test design

estimated d =X−μ2

s

So for the 1-sample t-test we estimate the Cohen’s d:

sample mean Pop mean from null hypothesis

sample standard deviation


Social Sciences Using SPSS: 1 sample t

• Entering the data– Observations go into one column

e.g., 2, 6, 5, 9

• Performing the analysis– Analyze -> Compare means -> one sample t-test

– Identify which column to do your test on

– Enter the ‘Test value’ – this is the population mean in the H0

– Be careful, the default is 0, this may not be what you want

• Reading the output– Mean of the sample, the computed-t, degrees of freedom,

p-value (Sig.) (also reminds you of your test value)


Social Sciences Next time

• Related samples t-tests

• Independent samples t-tests

Using t-tests Basic introduction and 1-sample t-tests Statistics for the Social Sciences Psychology...

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