INTRODUCTION TO T-TEST

Chapter 7

Hypothesis testing

• Critical region

• Region of rejection

• Defines “unlikely event” for H0

distribution

• Alpha ( )

• Probability value for critical region

• If set .05 = probability of result

occurred by chance only 5x out of 100

• Critical value (cv)

• Value of the statistic for alpha

• p-value

• Actual probability of result occurring

Review: z-test

• Hypothesis: Special training in reading skills will produce

change in scores.

• Reading skills for population: N(65, 15)

• Treatment group: N = 25; M = 70

• Is there evidence that training has an effect?

N

Mz

Review: z-test

• State hypotheses • H0: µ = 65

• H1: µ ≠ 65 (2-tailed)

• α = .05; so critical region = ?

• zcv = 1.96

• Calculate test statistic (z-test = ?)

• Make decision and state conclusion

• Fail to reject H0

• “No evidence that special training changed scores, z (n = 25) =

1.67, p = ns.”

67.1

25

15

6570

z

N

Mz

Review: z-test

• BUT if expect training to improve scores…

• State hypotheses

• H0: µ = 65

• H1: µ > 65 (1-tail)

• α = .05; so critical region zcv = 1.645

• Calculate test statistic

• Make decision and state conclusion

• Reject H0 : “Training significantly improved scores, z (n = 25) =

1.67, p < .05.”

67.1

25

15

6570

z

Errors

Type I

error

Correct

decision

Correct

decision

Type II

error

Actual situation

NO Effect Effect

H0 True H0 False

Reject H0

Retain H0

Experimenter’s

Decision

Conclude there was

an effect when there

actually wasn’t – the

risk of that is

Conclude there wasn’t

an effect when there

actually was an effect –

also called

Type I and Type II errors

• Type I: Incorrectly conclude significant difference • Conclude treatment has an effect but really doesn’t

• Type II: Miss a significant result • Conclude no effect of treatment when it really does

• Which is worse error to make?

• Examples: • Law:

• Type I: Jury says guilty when innocent

• Type II: Jury says innocent when guilty

• Medicine:

• Type I: Doctor says cancer present when isn’t

• Type II: Doctor says no cancer when it is there

• Answer: it depends!

Setting your alpha level

• Lower alpha to

minimize chance of

Type I error

• But, then increase

chance of Type II

error!

Concerns with Alpha

• All-or-none decision

• Reject or accept null hypothesis

• Alpha (criteria) is set arbitrarily

• Null hypothesis logic is artificial

• No such thing as “no effect”

• Doesn’t give size of effect

• p-value is chance of occurrence

• Cannot say “very significant”!

• Sample size changes p-value (probability of rejecting H0)

Statistical Power

• What is the probability of making the correct decision??

• If treatment effect truly exists either…

• We correctly detect the effect or…

• We fail to detect the effect (Type II error or )

• So, the probability of correctly detecting is 1 -

• Power: probability that test will correctly reject null hypothesis

(i.e. will detect effect)

• Power depends on:

• Size of effect

• Alpha level

• Sample size

-3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 3

Reject H0

Small

effect size

Large

effect size

Large

sample

size ->

small SD

Small

sample

size ->

large SD

Concerns with z-test

• Assumptions

• Must have a normal distribution

• Must know population mean and deviation

• Population mean (µ)

• Population standard deviation (σ)

• If don’t know above info or can’t make

assumptions need to use other statistics!

• Use “statistics” to make inferences

• Inferential statistics test: t-test

T-test: Estimates

• We’ve been using the z-score or z-test

formulas…

• Where M = standard error of the mean

• Instead…

• Use t-test

• Where M is a sample mean

• Where µ can be the population mean or

hypothetical H0 mean (i.e. chance or 0)

• Where sM = estimated standard error

N

MZ

M

MZ

XZ

OR

n

s2

n

s

Ms

Mt

OR

From z to t

• We use the “one-sample t-test” when we don’t

know .

• Use the sample variance instead of population SD

parameter ( )

• And, use hypothesized µ from the H0

n

Mz

n

s

Mt

2

N

s

Mt

OR

t-distribution

• “Student’s t-distribution”

similar in shape to N(0,1)

• Symmetric around 0

• Spread of t is greater

than N(0,1)

• As N increases, curve

approaches N(0,1)

• Different distributions

for “df” or n -1

• “Degrees of freedom”

• Table A.3 pp384

t table

• Is the t* (tcv) at

or higher than t

(statistic) given

df and α level

selected?

Finding the Critical t*

• Find the df in the left column

• Go across top to find the selected a level.

• Find the critical value, t*, for the df and a.

• If the absolute value of t is greater than or equal to t* then the test is significant at the a level.

• *Not all df are provided, so use the smaller df

(larger t*) for a more conservative estimate.

Eating behaviors

• “ATE”: positive attitude toward eating scale

• 3-point scale: -1 neg, 0 neutral, +1 positive

• 5 items: Eating desserts, snacks, etc.

• Minimum: -5; Maximum: +5

• Null hypothesis

• ATE = 0

• Alternative hypothesis

• ATE ≠ 0

• What values are needed to calculate the t-test?

• N

• Sample M and SD

t-test:

• N = 40 women

• ATE results: M = 0.525, SD = 2.16

• H0: ATE = 0

• Calculate t-test:

• 1.5?? Look up in t-table tcv

• Not every df so use df = 30:

• tcv = 2.042 = p = .05, 2-tailed

• Does not exceed critical value, so not significant

• Reporting a t-statistic: t(df) = value, p = value

• Women were found to have a neutral feeling toward eating, t(39) = 1.5, p = ns.

Ms

Mt

5.135.0

525.0

40

16.2

0525.0

t

Eating behaviors

• Women rated their feelings toward eating on a 3-point

scale.

• Participants’ average response (M = 0.5, SD = 2.16)

was not found to be significantly different from a neutral

rating of zero, t(39) = 1.5, p = ns.

• The results suggest that women do not have an overly

positive or negative feeling toward eating.

Confidence intervals

• CI: % confident that interval contains population

mean (µ)

• % is determined by researcher (e.g. 85, 90, 95%)

• Formula for z-test and t-test:

• CI = M +/- z*(σM)

• CI = M +/- t*(sM)

• Where z* or t* is the critical value

• Example:

• CI = 86 +/- 1.96(1.7) = 82.67 to 89.33

• 95% confident pop mean in this range

Spatial map study

Spatial Map Study

• % correct: academic, athletic, residence halls, social, administrative,

parking locations on campus

• H0 = 50% correct, H1 > 50% correct (1-tailed)

• Information: N = 11, M = 0.36, SD = 0.15

• df 10; for α=.05 (1 tail): t* or tcv= 1.812

(tcv = 3.169 for p = .0005 (1tail))

Participants remembered significantly less than 50% of the campus

map (M = .36), t(10) = -3.11, p < .01.

CI = M +/- t*(sM) = 0.36 +/- 1.812 (.045) = 0.68 to 0.28

95% confident that true mean falls within that range

N

s

Mt

045.

14.0

11

15.

50.36.

t t = -3.11

Lateralization in Perception of Emotion

• Two chimeric faces – which one is

happier (or younger example)?

• Emotion is processed in the right

hemisphere

• Dependent variable:

• Total score: -36 to +36

• where 0 = no lateralization

• What are the hypotheses?

• Null hypothesis: Total score = 0

• Alternative hypothesis?

Total score ≠ 0

t-test example

• Lateralization study (N = 173)

• H0: totscore = 0

• Totscore M = -11.7, SD = 16.6, S2 = 276

• t-test: -11.7 – 0 = -11.7 = - 9.28

sqrt(276/173) 1.26

• -9.28?? Look up tcv in table…

• Use 120: tcv = 2.576 = p = .005

• Participants show a significant lateralization to detect emotion using the right hemisphere, t(172) = 9.28, p < .005

• CI = M +/- t*(sM) = -11.7 +/- 2.576 (1.26) = -8.45 to -14.95

Lateralization results

• Participants demonstrated a statistically

significant lateralization effect, t(172) = -9.28,

p < .05. Emotion was more influenced by the

right hemisphere (M = -11.7, CI (.95) = -8.45

to -14.95), as opposed to what would be

expected by chance (M = 0).