This document discusses sample design and the t-test. It covers the sample design process which includes defining the population, sample frame, sample size, and sampling procedure. It also discusses probability and non-probability sampling techniques. The document then explains what a t-test is and how it can be used to test for differences between two group means. It covers the assumptions, procedures, hypotheses, and interpretation of t-test results.

1. SAMPLE DESIGNING AND T TEST
Presented by
Mansi Dua, Mansi Marwah, Megna Baid, Aashi Gupta

2. SAMPLE DESIGN
A sample design is the framework, or road map, that
serves as the basis for the selection of a survey sample
and affects many other important aspects of a survey
as well. In a broad context, survey researchers are
interested in obtaining some type of information
through a survey for some population, or universe, of
interest.

3. SAMPLE DESIGNING PROCESS
STEP 1 • DEFINE POPULATION
STEP 2 • DEFINING SAMPLE FRAME
STEP 3 • DETERMINE SAMPLE SIZE
STEP 4 • DETERMINE SAMPLE PROCEDURE
STEP5 • EXECUTE SAMPLING DESIGN

4. 1. DETERMING POPULATION
Defining the universe or population of interest is the
first step in any sample design. The universe can be
finite or infinite, depending on the number of items it
contains.
The target population should be defined in terms of :
 Element- from which the information is desired eg.
Respondent
 Sampling unit- it is an unit of element that is available
fro selection at some point
 Extent- it refers to the geographical boundaries
 Time- it is the time period under consideration

5. 2. Defining sample frame
Preparing the list of all the items within the
population of interest is the next step in the sample
design process. It is from this list, which is also called
as source list or sampling frame, that we draw our
sample. It is important to note that our sampling
frame should be highly representative of the
population of interest.

6. 3. Determine sample size
Determination of sample size is the next step to
follow. This is the most critical stage of the sample
design process because the sample size should not
be excessively large nor it should be too small. It is
desired that the sample size should be optimum and
it should be representative of the population and
should give reliable results. Population variance,
population size, parameters of interest, and
budgetary constraints are some of the factors that
impact the sample size.

7. 4. Determine sample procedure
There are many sampling techniques out of which the researchers has to choose the
one which gives lowest sampling error, given the sample size and budgetary
constraints.
There are lot of sampling techniques which are grouped into two categoriesas:
 Probability Sampling-This Sampling technique uses randomization to make sure that
every element of the population gets an equal chance to be part of the selected
sample.
 Non- Probability Sampling- This technique is more reliant on the researcher’sability
to select elements for a sample. Outcome of sampling might be biased and makes
difficult for all the elements of population to be part of the sample equally.

8. • Simple random sampling
• Stratified sampling
• Cluster sampling
• Systematic sampling
Probability
sampling
• Convenience Sampling
• Quota sampling
• Snowball sampling
• Judgmental sampling
Non
probability
sampling

9.  Simple Random Sampling-
Every element has an equal chance of getting selected
to be the part sample. It is used when we don’t have any
kind of prior information about the target population.
 Stratified Sampling -
This technique divides the elements of the population
into small subgroups (strata) based on the similarity in
such a way that the elements within the group are
homogeneous and heterogeneous among the other
subgroups formed. And then the elements are randomly
selected from each of these strata.
 Cluster Sampling-
Our entire population is divided into clusters or sections
and then the clusters are randomly selected. All the
elements of the cluster are used for sampling. Clusters
are identified using details such as age, sex, location etc.
 Systematic Clustering –
Here the selection of elements is systematic and not
random except the first element. Elements of a sample
are chosen at regular intervals of population. All the
elements are put together in a sequence first where each
element has the equal chance of being selected.

10. NON PROBABILITY SAMPLING
 Convenience Sampling
Here the samples are selected based on the availability. This method is used when
the availability of sample is rare and also costly. So based on the convenience
samples are selected.
 Judgmental sampling
This is based on the intention or the purpose of study. Only those elements will be
selected from the population which suits the best for the purpose of our study.
 Quota Sampling
This type of sampling depends of some pre-set standard. It selects the
representative sample from the population. Proportion of characteristics/ trait in
sample should be same as population. Elements are selected until exact proportions
of certain types of data is obtained or sufficient data in different categories is
collected.
 Snowball Sampling
This technique is used in the situations where the population is completely unknown
and rare. Therefore we will take the help from the first element which we select for
the population and ask him to recommend other elements who will fit the description
of the sample needed.

11. 5. EXECUTE SAMPLING DESIGN
The final step is to execute the design and draw
inferences and results from the sample in respect of
the set objective for sampling.

12. WHAT IS A T-TEST ?
 A t-test is a type of inferential statistic used to
determine if there is a significant difference between
the means of two groups, which may be related in
certain features. It is mostly used when the data sets,
like the data set recorded as the outcome from flipping
a coin 100 times, would follow a normal distribution
and may have unknown variances. A t-test is used as a
hypothesis testing tool, which allows testing of
an assumption applicable to a population.
 A t-test looks at the t-statistic, the t-distribution values,
and the degrees of freedom to determine the statistical
significance.

13. Calculating T-Tests
 T-Distribution Tables-The T-Distribution Table is available in one-
tail and two-tails formats. The former is used for assessing cases which have
a fixed value or range with a clear direction (positive or negative). For
instance, what is the probability of output value remaining below -3, or
getting more than seven when rolling a pair of dice? The latter is used for
range bound analysis, such as asking if the coordinatesfall between -2 and
+2
 T-Values- The t-value is a ratio of the difference between the mean of the
two sample sets and the variation that exists within the sample sets.
 Degrees of Freedom-Degrees of freedom refers to the values in a study
that has the freedomto vary and are essential for assessing the importance
and the validity of the null hypothesis. Computation of these values usually
depends upon the number of data records available in the sample set.

14. Hypothesis
There are two kinds of hypotheses for a one sample t-test, the null
hypothesis and the alternative hypothesis. The alternative hypothesis
assumes that some difference exists between the true mean (μ) and the
comparison value (m0),
whereas the null hypothesis assumes that no difference exists. The
purpose of the one sample t-test is to determine if the null hypothesis
should be rejected, given the sample data. The alternative hypothesis
can assume one of three
forms depending on the question being asked. If the goal is to measure
any difference,
regardless of direction, a two-tailed hypothesis is used. If the direction
of the difference between the sample mean and the comparison value
matters, either an upper-tailed or lower-tailed hypothesis is used.

15. The null hypothesis remains the same for each type of one sample t-test.
The hypotheses are formally defined below:
•• The null hypothesis (H0) assumes that the difference between the true
mean (μ) and the comparison value (m0) is equal to zero.
•• The two-tailed alternative hypothesis (H1) assumes that the difference
between the true mean (μ) and the comparison value (m0) is not equal to
zero.
•• The upper-tailed alternative hypothesis (H1) assumes that the true mean
(μ) of the sample is greater than the comparison value (m0).
•• The lower-tailed alternative hypothesis (H1) assumes that the true mean
(μ) of the sample is less than the comparison value (m0).

16. The mathematical representations of the null and alternative hypotheses
are defined below:
•H0: μ = m0
•H1: μ ≠ m0 (two-tailed)
•H1: μ > m0 (upper-tailed)
•H1: μ < m0 (lower-tailed)
Note. It is important to remember that hypotheses are never about data,
they are about the processes which produce the data. If you are interested
in knowing whether the mean weight of a sample of laptops is equal to
five pounds,
the real question being asked is whether the process that produced those
laptops has a mean of five.

17. ASSUMPTIONS
As a parametric procedure (a procedure which estimates
unknown parameters), the one sample t-test makes several
assumptions. Although t-tests are quite robust, it is good
practice to evaluate the degree of deviation from these
assumptions in order to assess the quality of the results.
The one sample t-test has four main assumptions:
 • The dependent variable must be continuous
(interval/ratio).
 • The observations are independent of one another.
 • The dependent variable should be approximately
normally distributed.
 • The dependent variable should not contain any
outliers.

18.  Level of Measurement
 The one sample t-test requires the sample data to be numeric and
continuous, as it is based on the normal distribution. Continuous data
can take on any value within a range (income, height, weight, etc.).
The opposite of continuous data is discrete data, which can only
take on a few values (Low, Medium, High, etc.). Occasionally,
discrete data can be used to approximate a continuous scale, such
as with Likert-type scales.
 Independence
 Independence of observations is usually not testable, but can be
reasonably assumed if the data collection process was random
without replacement. In our example, we would want to select laptop
computers at random, compared to using any systematic pattern.
This ensures minimal risk of collecting a biased sample that would
yield inaccurate results.

19.  Normality
 To test the assumption of normality, a variety of methods are available, but the
simplest is to inspect the data visually using a histogram or a Q-Q scatterplot. Real-
world data are almost never perfectly normal, so this assumption can be considered
reasonably met if the shape looks approximately symmetric and bell-shaped. The
data in the example figure below is approximately normally distributed.
 Outliers
 An outlier is a data value which is too extreme to belong in the distribution of
interest. Let’s suppose in our example that the assembly machine ran out of a
particular component, resulting in a laptop that was assembled at a much lower
weight. This is a condition that is outside of our question of interest, and therefore
we can remove that observation prior to conducting the analysis. However, just
because a value is extreme does not make it an outlier. Let’s suppose that our
laptop assembly machine occasionally produces laptops which weigh significantly
more or less than five pounds, our target value. In this case, these extreme values
are absolutely essential to the question we are asking and should not be removed.
Box-plots are useful for visualizing the variability in a sample, as well as locating
any outliers. The boxplot on the left shows a sample with no outliers. The boxplot on
the right shows a sample with one outlier.

20. PROCEDURE
The procedure for a one sample t-test can be summed up
in four steps. The symbols to be used are defined below:
 Y = Random sample
 yi = The ith observation in Y
 n = The sample size
 m0 = The hypothesized value
 y¯¯¯ = The sample mean
 σ^ = The sample standard deviation
 T =The critical value of a t-distribution with (n − 1)
degrees of freedom
 t = The t-statistic (t-test statistic) for a one sample t-test
 p = The p-value (probability value) for the t-statistic.

21. PROCEDURE
The four steps are listed below:
 1. Calculate the sample mean.
 y¯¯¯ = y1 + y2 + ⋯ + ynn
 2. Calculate the sample standard deviation.
 σ^ = (y1 − y¯¯¯)2 + (y2 − y¯¯¯)2 + ⋯ + (yn − y¯¯¯)2n −
1−−−−−−−−−−−−−−−−−−−−−−−√
 3. Calculate the test statistic.
 t = y¯¯¯ − m0σ^/n√
 4. Calculate the probability of observing the test statistic under the null hypothesis.
This value is obtained by comparing t to a t-distribution with (n − 1) degrees of
freedom. This can be done by looking up the value in a table, such as those found in
many statistical textbooks, or with statistical software for more accurate results.
 p = 2 ⋅ Pr(T > |t|) (two-tailed)
 p = Pr(T > t) (upper-tailed)
 p = Pr(T < t) (lower-tailed)
 Once the assumptions have been verified and the calculations are complete, all that
remains is to determine whether the results provide sufficient evidence to reject the
null hypothesis in favour of the alternative hypothesis.

22. INTERPRETATION
There are two types of significance to consider when interpreting the
results of a one sample t-test, statistical significance and practical
significance.
Statistical Significance
 Statistical significance is determined by looking at the p-value. The p-value gives the
probability of observing the test results under the null hypothesis. The lower the p-value, the
lower the probability of obtaining a result like the one that was observed if the null
hypothesis was true. Thus, a low p-value indicates decreased support for the null hypothesis.
However, the possibility that the null hypothesis is true and that we simply obtained a very
rare result can never be ruled out completely. The cutoff value for determining statistical
significance is ultimately decided on by the researcher, but usually a value of .05 or less is
chosen. This corresponds to a 5% (or less) chance of obtaining a result like the one that was
observed if the null hypothesis was true.
Practical Significance
 Practical significance depends on the subject matter. In general, a result is practically
significant if the size of the effect is large (or small) enough to be relevant to the research
questions being investigated. It is not uncommon, especially with large sample sizes, to
observe a result that is statistically significant but not practically significant. Returning to the
example of laptop weights, an average difference of .002 pounds might be statistically
significant. However, a difference this small is unlikely to be of any interest. In most cases,
both practical and statistical significance are required to draw meaningful conclusions.

23. THANK YOU