Introduction-to-Tests based on T-distribution.pptx

1. Introduction to T-
distribution
The t-distribution is a probability distribution used in statistical analysis
when the sample size is small or the population standard deviation is
unknown. It is commonly used for hypothesis testing and constructing
confidence intervals.
Sa by Shriram Kargaonkar

2. Definition of T-distribution
The T-distribution, also known as the Student's t-distribution, is a
probability distribution that arises when estimating the mean of a normally
distributed population with a small sample size. It is a bell-shaped curve
that is similar to the standard normal distribution, but with heavier tails,
indicating a greater probability of extreme values.
The T-distribution is used in statistical hypothesis testing and confidence
interval estimation when the population standard deviation is unknown. It
is particularly useful for making inferences about the mean of a small
sample drawn from a normally distributed population.

3. Properties of T-distribution
1. The T-distribution is a family of continuous probability distributions
that are symmetric, bell-shaped, and centered around 0.
2. The shape of the T-distribution is determined by its degrees of
freedom, which reflects the amount of information available in the
sample.
3. As the degrees of freedom increase, the T-distribution approaches
the standard normal distribution, making it useful for small sample
sizes.

4. Assumptions of T-distribution
The T-distribution makes several key assumptions that must be met for it
to be applicable. These include that the data is normally distributed, the
samples are independent, and the population variance is unknown.
Violations of these assumptions can lead to inaccurate results when using
T-tests for hypothesis testing. It's crucial to carefully evaluate the data
before selecting the appropriate statistical method.

5. Hypothesis testing with T-distribution
1
Sampling Distributions
When the sample size is small, the
normal distribution may not accurately
represent the population. The T-
distribution is used to account for the
increased uncertainty in small sample
estimates.
2 Hypothesis Formulation
T-distribution is commonly used to test
hypotheses about population means
when the variance is unknown. The
null and alternative hypotheses are
formulated based on the research
question.
3
Test Statistic Calculation
The T-statistic is calculated by dividing
the difference between the sample
mean and the hypothesized population
mean by the standard error of the
mean. This accounts for the
uncertainty in the sample.

6. One-sample T-test
The one-sample T-test is a statistical hypothesis
test used to determine if the mean of a
population is significantly different from a
specified value. It is commonly used when the
sample size is small and the population standard
deviation is unknown.
The test assumes the data follows a normal
distribution and calculates a T-statistic that is
compared to a T-distribution to determine the p-
value and make a decision about the hypothesis.

7. Two-sample T-test
1 Comparing Two Means
The two-sample T-test is used to
determine if the means of two independent
populations are significantly different.
2 Assumptions
The test assumes the two samples are
normally distributed and have equal
variances.
3 Hypothesis Testing
The null hypothesis is that the two
population means are equal, while the
alternative hypothesis is that they are
different.
4 Interpreting Results
The T-statistic and p-value are used to
determine if the difference between the
means is statistically significant.

8. Paired T-test
The paired T-test is a statistical method used to compare the means of two related samples or
measurements. This test is particularly useful when analyzing data where each observation in one
sample is paired with a corresponding observation in the other sample, such as before-and-after
measurements or data collected from the same individuals under different conditions.
3
Steps
The paired T-test involves three main steps:
calculating the difference between each pair of
observations, finding the mean of these
differences, and then testing whether the mean
difference is significantly different from zero.
95%
Confidence
The paired T-test allows researchers to determine
whether the observed difference between the two
groups is statistically significant, with a
confidence level typically set at 95%.
The key assumptions of the paired T-test are that the differences between the paired observations are
normally distributed and that the observations are independent. Violating these assumptions can affect
the validity of the test results.

9. Confidence intervals with T-distribution
Concept Overview
T-distribution is used to
construct confidence intervals
when the population standard
deviation is unknown. It
accounts for the uncertainty in
estimating the standard
deviation.
Calculation
The formula uses the t-statistic,
sample mean, sample size,
and a desired confidence level
to determine the range of
plausible values for the
population mean.
Interpretation
The confidence interval
represents the range of values
where the true population
mean is likely to fall. Wider
intervals indicate greater
uncertainty.

10. Applications and Examples
Real-World
Scenarios
T-distribution is
widely used in
various fields, such
as finance,
engineering, and
medical research, to
make inferences
about population
parameters when
sample sizes are
small.
Hypothesis
Testing
1. Evaluating the
effectiveness
of a new drug
treatment
2. Comparing
the average
income of
two different
regions
3. Assessing
the reliability
of a new
manufacturin
g process
Confidence
Intervals
T-distribution is used
to construct
confidence intervals
for population
means, proportions,
and variances,
providing a range of
plausible values for
the unknown
parameter.
Quality Control
In quality control
applications, T-
distribution is used
to analyze sample
data and make
decisions about the
process, such as
determining if a
production line is
operating within
acceptable limits.