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• Different versions of t-test
• Main usage of t-test
• t-test v/s z-test
• Assumptions of T-test
• Distribution of t-test and normal distribution
• Case Studies
• Introduced in 1908 by William Sealy Gosset for the quality control of
• Gosset published his mathematical work under the pseudonym
• It can be used to determine if two sets of data are significantly
different from each other, and is most commonly applied when the
test statistic would follow a normal distribution if the value of a
scaling term in the test statistic were known.
Different Version of T-test
• Single sample t – we have only 1 group; want to test against a
• 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.
Main Usage of t-test
Among the most frequently used t-tests are:
• A one-sample location test of whether the mean of a population has a
value specified in a null hypothesis.
• A two-sample location test of the null hypothesis that the means of two
populations are equal. All such tests are usually called Student's t-tests,
though strictly speaking that name should only be used if the variance of
the two populations are also assumed to be equal; the form of the test
used when this assumption is dropped is sometimes called Welch’s t-test.
These tests are often referred to as "unpaired" or "independent
samples" t-tests, as they are typically applied when the statistical units
underlying the two samples being compared are non-overlapping.
Main Usage of t-test
• A test of the null hypothesis that the difference between two responses
measured on the same statistical unit has a mean value of zero. For
example, suppose we measure the size of a cancer patient's tumor
before and after a treatment. If the treatment is effective, we expect the
tumor size for many of the patients to be smaller following the
treatment. This is often referred to as the "paired" or "repeated
measures" t-test: see paired difference test.
• A test of whether the slope of a regression line differs significantly
• We can use z-test to test hypotheses about means for large samples
( X X )2
( X )
H 0 : 10; H1 : 10; s X 5; N 200
X 11 z
2.83; 2.83 1.96 p 0.05
We use t-test when the sample size is small (N<100, the usual case) and the population
variance is unknown (the usual case).
Degrees of Freedom
The degrees of freedom for t-distribution 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.
• Based on assumptions of normality and homogeneity of variance.
• Tested using any statistical package.
• As long as the samples in each group are large and nearly equal, the ttest is robust, that is, still good, even thought assumptions are not met.
The t distribution is a short, fat relative of the normal. The shape of t depends on its df. As N becomes infinitely
large, t becomes normal.
One Sample t-test
• It is used in measuring whether a sample value significantly differs from a hypothesized value.
For example, a research scholar might hypothesize that on an average it takes 3 minutes for
people to drink a standard cup of coffee. He conducts an experiment and measures how long it
takes his subjects to drink a standard cup of coffee. The one sample t-test measures whether
the mean amount of time it took the experimental group to complete the task varies
significantly from the hypothesized 3 minutes value.
• It is used in comparing the means of two variables for a single group. This test computes the
differences between values of two variables for each case and tests whether the average differs
from 0. For example, in a study on impact of a particular diet on weight, all patients are
measured at the beginning of the study, prescribed a fixed diet, and measured again. Thus each
subject has two measures, often called before and after measures.
Independent Samples t-test
• The independent-Samples t-test procedure compares means for two groups of cases. Patients
with high blood pressure are randomly assigned to a placebo group and a treatment group. The
placebo subjects receive an inactive pill, and the treatment subjects receive a new drug that is
expected to lower blood pressure. The two-sample t test is used to compare the average blood
pressures for the placebo group and the treatment group.
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