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Small Sample Test T –Test &
Paired Sample Test
What is the t-Test?
t-Tests are a statistical way of testing a hypothesis
We do not know the population variance
The sample size is less than 30 (n<30).
We perform a One-Sample t-test when we want
to compare a sample mean with the population mean.
The difference from the Z Test is that we do not have the
information on Population Variance/SD here. We use
the sample standard deviation instead of population
standard deviation in this case.
We want to determine if on average girls score
more than 600 in the exam. We do not have the
information related to variance (or standard
deviation) for girls’ scores. To a perform t-test, we
randomly collect the data of 10 girls with their marks
and choose our ⍺ value (significance level) to be
0.05 for Hypothesis Testing.
In this example:
Mean Score for Girls is 606.8
The size of the sample is 10
The population mean is 600
Standard Deviation for the sample is 13.14
Our tcal value is less than the critical value & falls in the
Conclusion: Average girls score is less than or equal to
600 in the exam.
An experiment is conducted to determine
whether intensive tutoring is more effective than
paced tutoring. Two randomly chosen groups are
tutored separately and then administered proficiency
tests. Use a significance level of α < 0.05.
Null hypothesis: H0: μ 1 = μ 2 or H0: μ 1 – μ 2 = 0
Alternative hypothesis: Ha : μ 1 > μ 2 or Ha : μ 1 – μ 2 >
The degrees of freedom: This parameter is the
smaller of (n1 – 1) and (n2 – 1). Because this is a
one‐tailed test, the alpha level (0.05), DoF=9.
Look up t .05,9in the t‐table: 1.833
Since, tcal < tcritical, & the tcal value falls in the
acceptance region => the Null hypothesis is
Conclusion: This test has provided statistically
significant evidence that intensive tutoring is same
as of the paced tutoring.
Tcal = -2.44
Tcritical: alpha=0.05, dof=21, = 2.080
Reject Null Hypothesis.
Conclusion: The size of the tomato plants differ
Researchers at a pharmaceutical company want to test a
new pain-relief medicine. Specifically, they want to see if its
effect takes less time than their old medicine.
They take a sample of people who suffer from chronic
pain and randomly assign them into two groups. The first
group receive the old medicine while the second group receive
the new medicine (the participants don't know which medicine
Then, the participants are asked to measure the time
since they take the medicine until their pain is gone. Here is a
summary of the results:
Paired sample t-Test:
In paired sample hypothesis testing, a sample
from the population is chosen and two
measurements for each element in the sample are
Each set of measurements is considered a
sample. Unlike the hypothesis testing studied so far,
the two samples are not independent of one another.
Paired samples are also called matched
samples or repeated measures.
When to Choose a Paired T Test ?
Choose the paired t-test if you have two measurements
on the same item, person or thing. You should also choose
this test if you have two items that are being measured with a
For example, you might be measuring car safety
performance in Vehicle Research and Testing and subject the
cars to a series of crash tests. Although the manufacturers are
different, you might be subjecting them to the same conditions.
With a “regular” two sample t test, you’re comparing the
means for two different samples. For example, you might test
two different groups of customer service associates on a
business-related test or testing students from two universities
on their English skills. If you take a random sample each
group separately and they have different conditions, your
samples are independent and you should run an independent
samples t test
The null hypothesis for the independent samples
t-test is μ1 = μ2. In other words, it assumes the
means are equal. With the paired t test, the null
hypothesis is that the pairwise difference between
the two tests is equal (H0: µd = 0).
For example, if you want to determine whether
drinking a glass of wine or drinking a glass of beer
has the same or different impact on memory, one
approach is to take a sample of say 40 people, and
have half of them drink a glass of wine and the other
half drink a glass of beer, and then give each of the
40 people a memory test and compare results. This
is the approach with independent samples.
Another approach is to take a sample of 20
people and have each person drink a glass of wine
and take a memory test, and then have the same
people drink a glass of beer and again take a
memory test; finally we compare the results. This is
the approach used with paired samples.
A clinic provides a program to help their clients lose weight and
asks a consumer agency to investigate the effectiveness of the
program. The agency takes a sample of 15 people, weighing each
person in the sample before the program begins and three months
later to produce the table given below. Determine whether the
program is effective.
Calculate a paired t test by hand for the following data:
If you don’t have a specified alpha, use 0.05 (5%). For
this example problem, with df = 10 (n-1), the t-value is
Since, tcal > tcritical, reject Null Hypothesis.
Conclusion: there is difference between means of two
Twelve cars were equipped with radial tires and
driven over a test course. Then the same 12 cars
(with the same drivers) were equipped with regular
belted tires and driven over the same course. After
each run, the cars’ gas economy (in km/l) was
measured. Is there evidence that radial tires produce
better fuel economy? (Assume normality of data, and
use alpha = 5%)
The P value, or calculated probability, is the
probability of finding the observed, or more extreme,
results when the null hypothesis (H0) of a study
question is true.
In statistics, the p-value is the probability of
obtaining results at least as extreme as the observed
results of a statistical hypothesis test, assuming that
the null hypothesis is correct
Let, alpha = 5%
If p = 0.02, then p-value < alpha => Reject Ho
If p = 0.089 then p-value > alpha => Don’t Reject Ho
Calculating a p-value:
The p-value is calculated using the sampling distribution of the test
statistic under the null hypothesis, the sample data, and the type of test
being done (lower-tailed test, upper-tailed test, or two-sided test).
The p-value for:
a lower-tailed test is specified by:
p-value = P(TS <= ts | H0 is true) = cdf(ts)
an upper-tailed test is specified by:
p-value = P(TS >= ts | H0 is true) = 1 - cdf(ts)
assuming that the distribution of the test statistic under H0 is symmetric
about 0, a two-sided test is specified by:
p-value = 2 * P(TS > |ts| | H0 is true) = 2 * (1 - cdf( |ts| ) )
P -Probability of an event
TS -Test statistic
ts -observed value of the test statistic calculated from your sample
cdf() -Cumulative distribution function of the distribution of the test statistic
(TS) under the null hypothesis
Given: n=25, µ=260, X(bar)=330.6, SD(σ)=154.2
Z = x(bar) - µ / SDerror = 2.2922
= Area from µ = 0.4890 (from Z-table)
Thus, P-value = 2 x (0.5- 0.4890)
P-value = 0.022
Key Terms in Hypothesis Test:
Null hypothesis (H0):
The null hypothesis states that a population parameter (such
as the mean, the standard deviation, and so on) is equal to a
Alternative Hypothesis (H1):
The alternative hypothesis states that a population parameter
is smaller, greater, or different than the hypothesized value in
the null hypothesis.
Use a two-sided alternative hypothesis (also known as a non-
directional hypothesis) to determine whether the population
parameter is either greater than or less than the hypothesized
Use a one-sided alternative hypothesis (also known as a
directional hypothesis) to determine whether the population
parameter differs from the hypothesized value in a specific
A confidence interval is a range of values, derived from sample
statistics, that is likely to contain the value of an unknown population
The confidence level represents the percentage of intervals that
would include the population parameter if you took samples from the
same population again and again.
The single value estimates a population parameter by using your
The range of values estimates a population parameter by using your
A test statistic is a random variable that is calculated
from sample data and used in a hypothesis test. You can
use test statistics to determine whether to reject the null
hypothesis. The test statistic compares your data with
what is expected under the null hypothesis. The test
statistic is used to calculate the p-value.
A critical value is a point on the distribution of the
test statistic under the null hypothesis that defines a set
of values that call for rejecting the null hypothesis. This
set is called critical or rejection region.