The t test

The t-test
Inferences about Population Means

Properties of Student’s t
distribution
• Similar to Standard normal distribution
– Symmetric
– unimodal
– Centred at zero
• Larger spread about zero.
– The reason for this is the increased variability
introduced by replacing σ by s.
• As the sample size increases (degrees of
freedom increases) the t distribution
approaches the standard normal distribution

Background
• The t-test is used to test hypotheses
about means when the population
variance is unknown (the usual case).
Closely related to z, the unit normal.
• Developed by Gossett for the quality
control of beer.
• Comes in 3 varieties:
• Single sample, independent samples,
and dependent samples.

What kind of t is it?
• Single sample t – we have only 1 group; want
to test against a hypothetical mean.
• Independent samples t – we have 2 means, 2
groups; no relation between groups, e.g.,
people randomly assigned to a single group.
• Dependent t – we have two means. Either
same people in both groups, or people are
related, e.g., husband-wife, left hand-right
hand, hospital patient and visitor.

Degrees of Freedom
For the t distribution, degrees of freedom are always a
simple function of the sample size, e.g., (N-1).
One way of explaining df is that if we know the total or
mean, and all but one score, the last (N-1) score is not free
to vary. It is fixed by the other scores. 4+3+2+X = 10.
X=1.

1-Sample t-test
• Z test requires that you know σ from
pop
• Use a t-test when you don’t know the
population standard deviation.
• One sample t-test:
– Compare a sample mean to a population
with a known mean but an unknown
variance
– Use Sy (sample std dev) to estimate σ (pop
std dev)

The one-sample t-test (also called the
"single-parameter t test" or "single-sample
t-test") is used to determine whether a
sample comes from a population with a
specific mean. This population mean is not
always known, but is sometimes
hypothesized.

When to Use a One Sample T Test‐ ‐
To determine whether some obtained value is
statistically different from a neutral value, from a
previously published population mean, from zero,
or from some other externally dictated mean score,
a one-sample t-test can be used. The one-sample t-
test asks whether the mean score from the sample
you have tested is statistically different from the
externally determined mean score you are using to
compare it to.

Basic requirements of the one-sample t-test
• Assumption #1: You have one dependent
variable that is measured at the
continuous level.
• Assumption #2: The data are
independent (i.e., not correlated/related).

Study Designs
A one-sample t-test could be used as
part of a robust sampling strategy or
used when you want to make a
comparison to a criterion measure.

Sampling accuracy/adequacy
• The one-sample t-test can be used as
part of a robust sampling strategy. After
all, if the results from the sample are to
be generalized to the population, it is
important that the characteristics of the
sample reflect those of the population
that the sample is drawn from.

Comparison to a criterion measure
• The one-sample t-test can be used to
compare a value from a sample to a
criterion measure (i.e., to some other
value). The criterion measure may be
known or hypothesized.

Independent-samples t-test
The independent-samples t-test is used to
determine if a difference exists between
the means of two independent groups on a
continuous dependent variable. More
specifically, it will let you determine
whether the difference between these two
groups is statistically significant.

Basic requirements of the independent-samples
t-test
• Assumption #1: You have one dependent
variable that is measured at the continuous
level
• Assumption #2: You have one independent
variable that consists of two categorical,
independent groups (i.e., a dichotomous
variable). Example independent variables that
meet this criterion include gender (two groups:
"males" or "females"),

Study Designs
• An independent-samples t-test is most
often used to analyse the results of three
different types of study design: (a)
determining if there are differences
between two independent groups; (b)
determining if there are differences
between interventions; and (c)
determining if there are differences in
change scores.

Determining if there are differences between
two independent groups
• You have a study design where you are
measuring a dependent variable (e.g.,
weight, anxiety level, salary, reaction
time, etc.) in two independent/different
groups (e.g., males/females,
employed/unemployed, under 21 years
old/21 years old or older, etc.), and you
wish to know if there is a mean
difference in the dependent variable
between the two groups.

Determining if there are differences between
interventions
You have a study design where
participants are randomly assigned to one
of two groups. Each group receives a
different intervention (e.g., Group A
received no intervention, known as a
'control', whilst Group B undergo an
exercise programme; Group A listens to
music, whilst Group B does not listen to
music;

The paired-samples t-test
The paired-samples t-test is used to
determine whether the mean difference
between paired observations is
statistically significantly different from
zero. The participants are either the same
individuals tested at two time points or
under two different conditions on the
same dependent variable..

Basic requirements of the paired-samples t-test
Assumption #1: You have one dependent
variable that is measured at the
continuous (i.e., ratio or interval) level
Assumption #2: You have one
independent variable that consists of
two categorical, related groups or
matched pairs (i.e., a dichotomous
variable).

Study Designs
Determine if there are changes over time
between two related groups.
Example
Health professionals want to
investigate the medium-term effect of a
hypnotherapy programme on the daily
cigarette use of heavy smokers

Research question
Is there a difference in daily cigarette use amongst
heavy smokers six months after a hypnotherapy
programme?
Null hypothesis
There is no difference in daily cigarette use amongst
programme.
Alternative hypothesis
There is a difference in daily cigarette use amongst
programme.

Dependent variable
Cigarette consumption (measured in terms of the
number of cigarettes smoked daily)
Independent variable
Time, which has two levels:
Time point #1: Immediately before the start of the
hypnotherapy programme
Time point #2: 6 months after completion of the
hypnotherapy programme
The paired-samples t-test is used to determine
whether any difference between the two related
groups (i.e., the two time points) is statistically
significant.

Study Design #2
• Determine if there are differences between
conditions
• In this study design, you want to
determine if there are differences in the
scores of a dependent variable between
two related groups.

Example
An online retailer wants to know
whether using background music in
their order fulfilment centre would
lead to greater productivity amongst
packers

Research question
Is there a difference in productivity amongst
packers based on the use of background music?
Null hypothesis
There is no difference in productivity amongst
packers based on the use of background music.
Alternative hypothesis
There is a difference in productivity amongst
packers based on the use of background music.

Dependent variable
Productivity (measured in terms of packages
processed per hour)
Independent variable
Background music
Group 1: No Music (the "control")
Group 2: Music (the "treatment")
The paired-samples t-test is used to determine whether
any difference between the two related groups (i.e., the
two conditions) is statistically significant.

EXAMPLE
A researcher wants to test a new formula
for a sports drink that improves running
performance. Instead of a regular,
carbohydrate-only drink, this new sports
drink contains a new carbohydrate-
protein mixture. The researcher would
like to know whether this new
carbohydrate-protein drink leads to a
difference in performance compared to
the carbohydrate-only sports drink.

To do this, the researcher recruited 20 participants who
each performed two trials in which they had to run as far as
possible in 2 hours on a treadmill. In one of the trials they
drank the carbohydrate-only drink and in the other trial
they drank the carbohydrate-protein drink. The order of the
trials was counterbalanced and the distance they ran in both
trials was recorded.
For a paired-samples t-test, you will have two variables. In
this example, these are:
1) carb, which is the distance run (in km) in two hours for
the carbohydrate-only trial;
and
2) carb_protein, which is the distance run (in km) in two
hours for the carbohydrate-protein trial.

A behavioral scientist wants to know
whether drinking a single glass of beer
affects reaction times. She has 30
participants perform some tasks before
and after having a beer and records their
reaction times. For each participant she
calculates the average reaction time over
tasks both before and after the beer,
resulting in reaction_times.sav. Can we
conclude from these data that a single
beer affects reaction time?
EXAMPLE

The t test

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