applied statistics lecture_3

Introduction to applied statistics &

applied statistical methods

Prof. Dr. Chang Zhu1

objectives

• Going beyond the mean

• Tests of differences

� t-tests

� Wilcoxon and Mann Whitney tests

homogeneity of variance test

the Levene’s test

Analyse > Compare Means > One-way ANOVA

p > .05: equality of variance

p <.05 : no equality of variance

Test of Homogeneity of Variances

Levene

Statistic Df1 Df2 Sig.

3.766 1 56 .057

descriptive statistic

• measures of central tendency

(mean, median, mode)

• measures of spread or dispersion

(range, variance, standard deviation)

� show the characteristics of the sample, and the population as well


• How does the mean (M) and the standarddeviation (SD) tell us about thepopulation?


standard deviation between the sample means:

standard error (SE)

Central limit theorem (Field, 2009): The

sampling distribution (the frequency

distribution of sample means) has a

standard deviation calculated as:

s: standard deviation of the sample

n: the sample size


• In a normal distribution with a mean of 0 (zero) and

a standard deviation of 1, we can calculate the

probability of a score occurring.

�by looking at the probability table

(SPSS will do this for us)

z-scores

• We can convert a raw score into a z-score (a score in a normal distribution with a mean of 0, SD = 1):

Important:

95% of z-scores lie between -1.96 and 1.96

or 95% of the scores will be within the limit +/- 2 SD from the mean in a normal distribution.

z-scores

Convert a raw score into z-score, given that:

raw score: = 7

mean: = 4

std. deviation = 3

z = 2 � a person who gave a score of 7 is quite above than the average,having 2 standard deviations from the mean.

the confidence interval

Based on the SE, we can calculate the boundaries

within with the population mean will fall.

• upper bound: mean + 1.96*SE

• lower bound: mean – 1.96*SE

� confidence interval: a range of scores the

population mean will fall within

questions

• standard error (SE)

• z-score

• confidence interval (CI)

the confidence interval by error bars

� The two groups differ in their means and confidence

intervals, hence they are likely to come from different

populations.

Gender N Mean Variance Std. Deviation

Stress at the

start of the

week

Female 16 14.81 28.16 5.307

Male 16 18.94 63.26 7.954

?

tests of differences

independent samples test

SPSS output

Analyse>Compare Means > Independent-Samples Test

t-test for Equality of Means

t df

Sig.

(2-tailed)

Mean

Difference

Std. Error

Difference

95% Confidence

Interval of the

Difference

Lower Upper

-1.726 30 .095 -4.125 2.390 -9.007 .757

-1.726 26.146 .096 -4.125 2.390 -9.037 .787

t = mean difference/Std. Error difference

independent samples t-tests

1. Do male and female participants have

different levels of computer use?

2. Does age group 1 have a higher level of

computer use than age group 2?

� What do we need to find out?

� How many groups involved in the study?

15

paired samples t-tests

• Are the mean exam score in September

different to their exam score in November?

• Are the deep learning approach of students

reported at the beginning of the study

significantly different to the deep learning

approach of students reported at the end of

the study?

� What do we need to find out?

� How many groups involved in the study?

16

test of differences

The t-test assesses whether the means of two

groups are statistically different from each other.

Assumptions:

• normal distribution

• homogeneity of variance (equal variances)

• data measured at interval level (scale)

violated? � non-parametric equivalent tests

nonparametric tests of differences

A group of students were asked to rank the extent to which they fear of

statistics from 1 (scared) to 7 (not scared at all) at 2 different times

time 1 1 3 5 2 7 3 3 4 6

time 2 5 2 5 2 6 1 7 6 2

difference in ranks -4 1 0 0 1 2 -4 -2 4

ties positive negative

parametric tests nonparametric tests objectives

the single sample t-test whether the observed

mean is different from a

set value

the independent t-test the Wilcoxon rank-sum

test and the Mann

Whitney test

comparing means from

two independent groups of

individuals

the paired t-test the Wilcoxon signed

rank test

comparing the means of

two sets of observations

from the same individuals

or from pairs of individuals

tests of differences

Mean (SD) Mean

difference

t Sig.

score or

measurement at

time point 1

score or

measurement at

time point 2

Paired sample t-test results

reporting the results

reporting the results(APA styles)

PRACTICE

what tests to use?

A TV company have started a reality TV show where 32 members of the public are

left to fend for themselves on a desert island. They have asked a psychologist to

monitor the psychological well-being of the contestants and he records a number of

indices of mental health. He is initially interested in the amount of stress

experienced by the contestants during their first week on the island and

hypothesises that:

(1)the females will report higher levels of stress than the males at the start as well

as at the end of the week (H1)

(2)the level of stress experienced by all the participants is increased by the end of

the week of the reality TV show (H2)

The data is named TVshow.sav

H1: Analyze > Compare Means > Independent-Samples Test

H2: Analyze > Compare Means > Paired-Samples T-Test

what tests to use?

We want to know if people who intend to get a Ph.D. or Psychology Doctor (PhD

holder) in psychology are more likely to rely on a calendar or day-planner to

remember what they are supposed to be doing (i.e., are people who might become

professors more absent minded than other people).

The ordinal variable planner measures the extent to which a person relies on a

calendar/day planner, ranging from 1 (strongly agree) to 5 strongly disagree).

The data file is named planner_use.sav.

Analyse > Nonparametric Tests > Legacy Dialogs >

2 Independent-Samples

Practice 1

- use the data file TVshow.sav

• Did females experience a higher level of stress at the

start of the week than males?

• Did females experience a higher level of stress at the

end of the week than males?

Which test should you use?

independent t-test

� Practical guidelines page 2Analyze > Compare Means > Independent-Samples Test

Practice 1

Independent Samples Test

Levene's Test for

Equality of

Variances t-test for Equality of Means

F Sig. t df

Sig.

(2-tailed)

Mean

Difference

Std. Error

Difference

95% Confidence

Interval of the

Difference

Lower Upper

Stress at the start of the

week

Equal variances assumed 3.211 .083 -1.726 30 .095 -4.125 2.390 -9.007 .757

Equal variances not

assumed

-1.726 26.146 .096 -4.125 2.390 -9.037 .787

Stress at the end of the

week

Equal variances assumed .038 .847 .522 30 .606 2.188 4.193 -6.376 10.751

Equal variances not

assumed

.522 29.673 .606 2.188 4.193 -6.380 10.755

Analyse>Compare Means > Independent-Samples Test

Practice 1

Conclusion?

At the start of the week, the male participants experienced a

higher level of stress (M= 18.94, SE = 1.99) than the females

(M=14.81, SE = 1.32). This difference was significant t(30) = -

1.73, p < .05. Therefore, hypothesis 1 is not supportedbecause the psychologist assumed that the females

experienced a higher level of stress than males.

(Practical guidelines page 3)

Practice 2

- use the data file planner_use.sav

• Do people who intend to do a PhD degree or PhD

holders are more likely to use a calendar or day

planner?

Which test to use?

independent t-test (nonparametric)

� Practical guidelines page 6



Practice 2

Ranks

Intend To Get PhD or PsyD N Mean Rank Sum of Ranks

I rely on a calendar / day-planner

to remember what I am supposed

to do.

Intend to do a PhD 11 27.32 300.50

PhD holder 35 22.30 780.50

Total 46



Practice 2



Test Statisticsb

I rely on a calendar / day-planner to remember what I am

supposed to do.

Mann-Whitney U 150.500

Wilcoxon W 780.500

Z -1.169

Asymp. Sig. (2-tailed) .242

Exact Sig. [2*(1-tailed

Sig.)]

.284a

Exact Sig. (2-tailed) .252

Exact Sig. (1-tailed) .127

Point Probability .006

a. Not corrected for ties.

b. Grouping Variable: Intend To Get PhD or PsyD

Practice 2

Conclusion?

People who intend to do a PhD do not differ significantly from

PhD degree holders with regard to the use of day planner to

remember what they are supposed to be doing , U = 150.50, z =

-1.169, p > .05, ns.

(Practical guidelines page 7)

Assignment 3

•

Lecture 3_practical guidelines_assignment(p. 10)

applied statistics lecture_3

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