Revision:Descriptive Statistics

and the t-test2009 Methodology A - Lecture 2

1. Today’s Learning Objectives

2. Descriptive Statistics

a. Central Tendency

b. Variability

3. Hypothesis Testing

4. The t-test

a. Degrees of freedom

b. Effect Size

5. Review of Learning Objectives

6. Vocabulary

Outline Today’s Learning ObjectivesDescriptive Statistics

1. What are the three most common measures of central tendency?

2. How do you calculate the mean?3. How do you calculate the median?4. How do you calculate the mode?5. What are the two measures of

variability?6. How are these two measures

related?

Hypothesis Testing7. What is the null hypothesis?8. What is the difference between

one-tailed and two-tailed alternative hypotheses?

9. How do p-values relate to the null and alternative hypotheses?

10. How do Type I and Type II errors differ?

The t-test11. What are the three types of t-test?12. What do you need to know about

your data to compute the t statistic?

13. What are the assumptions of the t-test?

14. How do you test for equal variance?

15. What do you do if variance is not equal?

16. What are degrees of freedom?17. How do you calculate effect size?18. How do you report the outcome of

a t-test?19. Given sample data, which type of

t-test is most appropriate?

Descriptive Statistics

Ch. 3

Central Tendency

Mean = (2+3+4+4+5+6+7+9)/8 = 5

Median = (4+5)/2 = 4.5

Mode = 4

2, 3, 4, 4, 5, 6, 7, 9

41-6

Variability

Variance = !(x-x)2 / (n-1)

Standard Deviation = "variance

x = 2, 3, 4, 4, 5, 6, 7, 9

74-7

x

5 6 7 8 94321

Variability

x = 2, 3, 4, 4, 5, 6, 7, 92 3 4 4 6 7 9

Variability

x = 2, 3, 4, 4, 5, 6, 7, 9

= 36 / 7 5.14=

363534333231302928

272625242322212019

181716151413121110

987654321

!(x-x)2 / (n-1)

Variability

x = 2, 3, 4, 4, 5, 6, 7, 9

5.14SD = " 5.14 = 2.27

363534333231302928

272625242322212019

181716151413121110

987654321

! Null Hypothesis (H0)

! Alternative Hypothesis (HA or Hi)

! Statistical significance (p-value)

! Type I Error

! Type II Error

Hypothesis Testing

Ch. 5

Null Hypothesis (H0)

! There is no difference between groups (paired-samples and independent-samples t-tests)

! The mean is not different from a specified value (one-sample t-test)

Alternative Hypothesis (HA)

! There is no difference between groups (paired-samples and independent-samples t-tests)

! The mean is not different from a specified value (one-sample t-test)

Statistical Significance! Gives guidelines for what allows us to

accept or reject the null hypothesis

! The p-value is the probability that you would get your test statistic (e.g. t) if the null hypothesis were true

! Alpha (#) is the critical p-value:

! accept H0 when p $ #! reject H0 when p < #

! In psychological research, usually # = .05

Tails! Two-tailed hypothesis: HA does not specify

in what direction the difference lies

! One-tailed hypothesis: the direction is specified

! One-tailed hypotheses require a convincing argument that the direction was predicted before the data were collected

! SPSS reports 2-tailed p-values for t-tests; since the t distribution is symmetric, divide p by 2 for the 1-tailed p-value

150-4

Types I and II ErrorsType I error:

! rejecting H0 when H0 is actually true

! a false positive

! p-value is ‘too small’

! e.g. a pregnancy test says you are pregnant when you are not

Type II error:

! accepting H0 when H0 is actually false

! a false negative

! p-value is ‘too big’

! e.g. a pregnancy test says you are not pregnant when you are

146-9

1. Independent-samples

2. Paired-samples

3. One-sample

The t-test

Ch. 7

Data are approximately normally distributed

For independent-samples t-tests:

Variance is equal or use a correction for

unequal variance

Assumptions

222-3

If Levene’s test is significant (p < .05), use ‘Equal variances not assumed’ (round df to the nearest whole number). If p $ .05, use ‘Equal variances assumed’.

One score from each subject and two groups of subjects (e.g. male and female)

Independent-samples t-test

Are the two groups of subjects doing something different?

223-30

Independent-samples t-test

tdifference between group means

variability of the groups=Two scores from each subject

Tests if the difference between scores is greater than 0

Paired-samples t-test

Are subjects doing something different in two conditions?

230-5

Paired-samples t-test

tdifference between paired scores

variability of the difference scores=

0

One score from each subject and a strong hypothesis about what the mean should be if H0 is true, usually 0 or chance (e.g. 50% in a forced-choice task)

One-sample t-test

Are scores different from chance?

One-sample t-test

tdifference between mean and chance

variability of the scores=

0

Independent-samples t-test

df = (N1 - 1) + (N2 - 1) = Ntotal - 2

Paired-samples t-test

df = N - 1

One-sample t-test

df = N - 1

Degrees of Freedom (df)

221

N - 1

Degrees of Freedom (df)

221

(N1 - 1) + (N2 - 1)

Degrees of Freedom (df)

221

Effect Size

ddifference between group means

standard deviation=

Effect SizePaired-samples t-test

= = 0.87d = 8.353

9.617StDev

Mean

Effect SizeOne-sample t-test

= = 3.5426.322

7.430d =Mean Difference

Std. Deviation

Effect Size

d =abs( Mean1 - Mean2 )

( StDev1 + StDev2 ) / 2

Independent-samples t-test

( 7.689 + 7.209 ) / 2 = 0.35d =abs( 24.42 - 27.04 )

Effect Size

How big is the difference between the groups, compared to the variability within the groups?

dsmall = 0.2 dmedium = 0.5 dlarge = 0.8

alpha (#)alternative hypothesis (HA or Hi)central tendencyCohen’s ddegrees of freedom (df)dependent-samples t-testdescriptiveeffect sizeindependent-samples t-testinferentialmeanmedianmode

null hypothesis (H0)one-sample t-testp-valuepaired-samples t-testrelated-samples t-teststandard deviationstatistical significancet statistict-testType I errorType II errorvariabilityvariance

Vocabulary