Type 1 and Type 2 errors are concepts from hypothesis testing

in statistics.

Type 1 Error (False Positive)

 Definition: A Type 1 error occurs when the null

hypothesis (H0H_0H0) is rejected when it is actually true.
 Consequence: This means you conclude that there is an
effect or a difference when there isn't one.
 Probability: The probability of making a Type 1 error is
denoted by α\alpha, which is also known as the
significance level of the test (commonly set at 0.05).

Type 2 Error (False Negative)

 Definition: A Type 2 error occurs when the null

hypothesis (H0H_0H0) is not rejected when it is actually
false.
 Consequence: This means you conclude that there is no
effect or difference when there actually is one.
 Probability: The probability of making a Type 2 error is
denoted by β\beta.

Relationship Between Type 1 and Type 2 Errors

 Inverse Relationship: Reducing the probability of a Type

1 error (α\alpha) typically increases the probability of a
Type 2 error (β\beta), and vice versa.
 Power of the Test: The power of a test is 1−β1 - \
beta1−β, representing the probability of correctly
 rejecting the null hypothesis when it is false. A higher
power means a lower probability of a Type 2 error.

Example

Suppose you are testing a new drug to see if it is more effective

than the current standard.

 Type 1 Error: Concluding that the new drug is more

effective when it actually is not.
 Type 2 Error: Concluding that the new drug is not more
effective when it actually is.

Understanding these errors is crucial for designing experiments

and interpreting results, ensuring that conclusions drawn are as
accurate and reliable as possible.

Parametric and non-parametric tests are two broad categories

of statistical tests used to infer properties of populations based
on sample data. Here’s a detailed comparison of the two:

Parametric Tests

Definition: Parametric tests assume that the data follows a

certain distribution, typically a normal distribution.

Key Characteristics:

1. Assumptions:
o The data must follow a specific distribution (usually
normal distribution).
 o Homogeneity of variance (equal variances within
groups).
o The data should be measured at least on an interval
scale.
2. Examples:
o t-tests (independent t-test, paired t-test)
o ANOVA (Analysis of Variance)
o Pearson’s correlation
o Linear regression
3. Advantages:
o More powerful than non-parametric tests if
assumptions are met.
o Can provide more precise estimates because they
use all the data and the parametric form of the data
distribution.
4. Disadvantages:
o Not suitable for data that doesn’t meet the
assumptions (e.g., non-normal distribution, ordinal
data).

Non-Parametric Tests

Definition: Non-parametric tests do not assume a specific

distribution for the data.

Key Characteristics:

1. Assumptions:
o Do not require the assumption of normality.
o Can be used on ordinal data or non-continuous data.
 o Often assume randomness and independence of
samples.
2. Examples:
o Mann-Whitney U test (alternative to the
independent t-test)
o Wilcoxon signed-rank test (alternative to the
paired t-test)
o Kruskal-Wallis test (alternative to one-way ANOVA)
o Spearman’s rank correlation
o Chi-square test
3. Advantages:
o More flexible as they don’t require the data to fit a
specific distribution.
o Can be used with small sample sizes.
o Suitable for ordinal data or data that does not meet
the assumptions of parametric tests.
4. Disadvantages:
o Generally less powerful than parametric tests if the
parametric test assumptions are met.
o Often require larger sample sizes to achieve the
same power as parametric tests.
o May provide less precise estimates.

Summary
  Parametric tests: Assume data follows a certain
distribution, more powerful if assumptions are met,
examples include t-tests and ANOVA.
 Non-parametric tests: Do not assume a specific
distribution, more flexible, can be used with non-normal
data and small sample sizes, examples include Mann-
Whitney U test and Wilcoxon signed-rank test.

Choosing between parametric and non-parametric tests

depends on the nature of your data and whether it meets the
assumptions required for parametric tests.

What are degrees of freedom in statistical model?

In statistical modeling, "degrees of freedom" (often abbreviated

as "df") refers to the number of independent values or
quantities that can vary in an analysis without violating any
constraints. Degrees of freedom are crucial in many statistical
tests and procedures because they help determine the
distribution of test statistics under the null hypothesis.

Here are some key points about degrees of freedom in different

contexts:

1. Simple Example: Suppose you have three numbers

whose sum must be 15. If you know the values of two
numbers, the third number is automatically determined.
Thus, you have 3−1=23 - 1 = 23−1=2 degrees of
freedom because you can freely choose any values for two
of the numbers.
2. In Linear Regression:
 o The total number of degrees of freedom is typically
the number of observations minus one.
o The residual degrees of freedom, which are used in
calculating the variance of residuals, is the total
number of observations minus the number of
estimated parameters (including the intercept).
3. In Hypothesis Testing:
o In a t-test comparing the means of two groups, the
degrees of freedom are often n1+n2−2n_1 + n_2 -
2n1+n2−2, where n1n_1n1 and n2n_2n2 are the
sample sizes of the two groups.
o In a chi-square test, the degrees of freedom are
usually the number of categories minus one (for a
goodness-of-fit test) or the product of the numbers of
categories minus one in each dimension (for a test of
independence).
4. ANOVA (Analysis of Variance):
o The total degrees of freedom are the total number of
observations minus one.
o The degrees of freedom for the model are the
number of groups minus one.
o The degrees of freedom for the error term are the
total degrees of freedom minus the degrees of
freedom for the model.

Understanding degrees of freedom is essential for interpreting

the results of statistical tests correctly, as they are used to
determine critical values from statistical distributions (such as
the t-distribution or chi-square distribution) and thus influence
p-values and confidence intervals.

Write the assumptions of t test

The t-test is a statistical test used to compare the means of two

groups and determine if they are statistically different from
each other. There are several types of t-tests, including the
one-sample t-test, independent two-sample t-test, and paired
sample t-test. Each type has specific assumptions that must be
met for the test to be valid. Here are the general assumptions
for the t-test:

1. Independence:
o Independent Two-Sample t-test: The
observations in each group must be independent of
each other. This means the samples from the two
groups should be randomly and independently drawn
from the population.
o Paired Sample t-test: The differences between
paired observations should be independent of each
other.
o One-Sample t-test: The observations must be
independent of each other.
2. Normality:
o The data in each group should be approximately
normally distributed. This assumption is more critical
when the sample size is small (typically less than 30).
For larger sample sizes, the Central Limit Theorem
states that the sampling distribution of the mean will
be approximately normally distributed, even if the
data are not.
3. Homogeneity of Variance (for independent two-sample
t-test):
o The variances of the two groups being compared
should be approximately equal. This is known as the
assumption of homogeneity of variance or
homoscedasticity. When this assumption is violated,
 a variation of the t-test called Welch's t-test can be
used, which does not assume equal variances.
4. Scale of Measurement:
o The dependent variable should be measured at the
interval or ratio level. This means the data should be
continuous and have a meaningful zero point and
equal intervals between values.
5. Random Sampling:
o The data should be obtained using a random
sampling method to ensure the sample is
representative of the population.

Meeting these assumptions helps ensure the validity of the t-

test results. If the assumptions are violated, the test may
produce misleading results, and alternative methods or
corrections may be necessary.

Differentiate between F test and t test

The F-test and t-test are both statistical tests used to analyze
the variability within datasets, but they serve different
purposes and are used in different contexts. Here’s a
breakdown of the key differences between the two:

Purpose

 t-test: Used to compare the means of two groups to

determine if they are significantly different from each
other.
  F-test: Used to compare the variances of two or more
groups to determine if they are significantly different. It is
also used in the context of comparing multiple means
(ANOVA).

Types

 t-test:
o Independent t-test: Compares the means of two
independent groups.
o Paired t-test: Compares the means of two related
groups (e.g., the same group before and after a
treatment).
o One-sample t-test: Compares the mean of a single
group to a known value or population mean.
 F-test:
o One-way ANOVA (Analysis of Variance):
Compares the means of three or more independent
groups.
o Two-way ANOVA: Compares the means with two
independent variables.
o F-test for equality of variances: Compares the
variances of two groups.

Distribution

 t-test: Based on the t-distribution, which is used when the

sample size is small and the population standard deviation
is unknown.
  F-test: Based on the F-distribution, which is a ratio of two
chi-square distributions. The F-distribution is used to
compare variances and to test hypotheses involving
multiple group means.

Assumptions

 t-test:
o The data should be approximately normally
distributed (especially for small sample sizes).
o The samples should have equal variances (though
Welch’s t-test can be used if this assumption is
violated).
o The data should be independent.
 F-test:
o The data should be normally distributed.
o The samples should have equal variances
(homogeneity of variance).
o The data should be independent.
Examples of Use

 t-test: Comparing the average test scores of two different

classes.
 F-test: Comparing the average test scores of students
from three or more different classes.

Summary

 Use a t-test when comparing the means of two groups.

 Use an F-test when comparing variances or when
comparing the means of three or more groups (ANOVA).

By understanding the differences between the F-test and t-test,

you can select the appropriate test for your specific data
analysis needs.

Write the assumptions of F test

The F-test is a statistical test used to compare the variances of

two or more groups to determine if they are significantly
different from each other. The assumptions of the F-test are as
follows:

1. Independence: The samples must be independent of

each other. This means that the data collected from one
group should not influence the data collected from
another group.
 2. Normality: The data in each group should be
approximately normally distributed. This assumption is
particularly important for small sample sizes. For larger
samples, the F-test is robust to moderate deviations from
normality.
3. Homogeneity of Variance (Homoscedasticity): The
variances of the populations from which the samples are
drawn should be equal. This means that the variability in
each group should be similar. If the variances are
significantly different, the results of the F-test may not be
valid.
4. Random Sampling: The data should be collected using a
random sampling method to ensure that the sample is
representative of the population.
5. Scale of Measurement: The dependent variable should
be measured on at least an interval scale, which means
the data should be continuous and have a meaningful
order and equal intervals.

If these assumptions are violated, the results of the F-test may

be unreliable. In such cases, alternative statistical methods,
such as the Welch’s ANOVA or non-parametric tests, may be
more appropriate.

Sampling is the process of selecting a subset of individuals or

observations from a larger population to estimate
characteristics of the whole population. It is a fundamental
technique used in statistics, survey research, and various
scientific disciplines to gather and analyze data without
needing to collect information from every member of the
population.

Explain different types of Statistics.

Write the steps of statistical inference
in details

Statistics is the science of collecting, analyzing, interpreting,

presenting, and organizing data. It can be broadly categorized
into two main types:

1. Descriptive Statistics

Descriptive statistics summarize and describe the features of a

dataset. This type of statistics provides simple summaries
about the sample and the measures. Some common techniques
and measures include:

 Measures of Central Tendency: Mean, median, mode

 Measures of Variability: Range, variance, standard
deviation
 Measures of Shape: Skewness, kurtosis
 Graphical Representations: Histograms, bar charts, box
plots, scatter plots

2. Inferential Statistics

Inferential statistics use a random sample of data taken from a

population to make inferences about the population. It involves
generalizing from a sample to a population, making predictions,
and testing hypotheses. Some common techniques include:

 Hypothesis Testing: T-tests, chi-square tests, ANOVA

 Confidence Intervals: Estimating the range within which
a population parameter lies
 Regression Analysis: Predicting the value of a
dependent variable based on the value of one or more
independent variables

Steps of Statistical Inference

Statistical inference is the process of using data analysis to

deduce properties of an underlying probability distribution.
Here are the detailed steps involved:

1. Define the Population and Sample

 Population: The entire set of individuals or items that we

are interested in studying.
 Sample: A subset of the population that is selected for
the actual study.

2. Formulate the Hypotheses

 Null Hypothesis (H0): A statement of no effect or no

difference, which we aim to test against.
 Alternative Hypothesis (H1 or Ha): A statement that
indicates the presence of an effect or difference.

3. Choose the Significance Level (α)

  The significance level is the probability of rejecting the
null hypothesis when it is actually true. Common choices
are 0.05, 0.01, and 0.10.

4. Select the Appropriate Test Statistic

 Depending on the nature of the data and the hypotheses,

select an appropriate statistical test (e.g., t-test, chi-
square test, ANOVA, regression analysis).

5. Collect Data and Calculate the Test Statistic

 Gather the data from the sample and calculate the value
of the test statistic.

6. Determine the P-value or Critical Value

 P-value: The probability of obtaining a test statistic at

least as extreme as the one observed, assuming the null
hypothesis is true.
 Critical Value: A threshold value that the test statistic
must exceed to reject the null hypothesis at the chosen
significance level.

8. Draw Conclusions
  Interpret the results in the context of the research
question. If the null hypothesis was rejected, it suggests
evidence in favor of the alternative hypothesis. If not, it
suggests insufficient evidence to support the alternative
hypothesis.

9. Report the Results

 Present the findings in a clear and concise manner,

including the test statistic, p-value, confidence intervals,
and any relevant graphs or charts.

By following these steps, researchers can make informed

decisions and draw valid conclusions from their data.