Hypothesis Testing.pptx

MATHEMATICAL
STATISTICS
PARAMETRIC & NON-PARAMETRIC
TEST
Presented To :
Poonam
Assistant Professor
Dept of Distance Education
Presented By :
Rishabh Jain
220101080001
MBA ( Business
Analytics )

HYPOTHESIS
TESTING
Hypothesis Testing refers to
1. Making an assumption , called hypothesis , about a population
parameter .
2. Collecting sample data
3. Calculating sample statistics
4. Using the sample statistics to evaluate hypothesis ( how likely it is
that our hypothesized parameter is correct . To test the validity of our
assumption we determine the difference between the hypothesized
parameter value and sample value . )

NULL
HYPOTHESIS
 The basic assumption regarding population parameter which
can be tested is called null hypothesis .
 In other words the statement that may difference between
observed sample statistics and specified population parameter
is due to a sampling error called null hypothesis .
 Therefore the null hypothesis means hypothesis of no difference
and it is denoted by Ho .

ALTERNATE
HYPOTHESIS
• When the null hypothesis is rejected than the
assumption taken as true is called alternate
hypothesis .
• It is denoted by Ha .
• The alternative hypothesis is a statement used in statistical
inference experiment. It is contradictory to the null hypothesis .

STEPS OF TESTING OF
HYPOTHESIS
1.State the null and alternate hypothesis
.
2.Choose the level of significance at size
α .
3.Determine the critical region
4.Use test statistics
5.Making decision or conclusion

PARAMETRIC
TEST
• The basic principle behind the parametric tests is that we have a fixed set of
parameters that are used to determine a probabilistic model .
• Parametric tests are those tests for which we have prior knowledge of the
population distribution .
• The parameters used in these test includes mean, standard deviation &
variance.
• The different parametric test are :
 T-test
 F-test
 Z-test
 Anova test

T-
TEST
• It is a parametric test of hypothesis testing based on Student’s T distribution.
• It is essentially, testing the significance of the difference of the mean values when the sample
size is small (i.e, less than 30)
Assumptions :
•Population distribution is normal, and
•Samples are random and independent
•The sample size is small.
•Population standard deviation is not known.
One Sample T-test: To compare a sample mean
with that of the population mean.
Two-Sample T-test: To compare the
means of two different samples.
t=
𝑥1−𝑥2
𝑠1
2
𝑁1
+
𝑠2
2
𝑁2
𝑧 =
𝑥 − 𝑢
𝜎 𝑛

Z-
TEST
1. It is a parametric test of hypothesis testing.
2. It is used to determine whether the means are different when the population variance is
known and the sample size is large ( i.e. , greater than 30).
Assumptions :
•Population distribution is normal
•Samples are random and independent.
•The sample size is large.
•Population standard deviation is known.
One Sample Z-test : To compare a sample mean
with that of the population mean.
Two Sample Z-test : To compare the means
of two different samples.
𝑧 =
𝑥1 + 𝑥2
𝜎1
2
𝑛1
+
𝜎2
2
𝑛2
𝑧 =
𝑥 − 𝑢
𝜎 𝑛

F-
TEST
1. It is a parametric test of hypothesis testing based on Snedecor F-distribution.
2. It is a test for the null hypothesis that two normal populations have the same variance.
3. An F-test is regarded as a comparison of equality of sample variances.
4. F-statistic is simply a ratio of two variances.
5. It is calculated :
F = s1
2/s2
2
𝑠2 = 𝑖=1
𝑛
𝑥𝑖 − 𝑥 2
𝑛 − 1
Assumptions :
•Samples are drawn randomly and independently.

ANOV
A
1. Also called as Analysis of variance, it is a parametric test of hypothesis testing.
2. It is an extension of the T-Test and Z-test.
3. It is used to test the significance of the differences in the mean values among more than two
sample groups.
4. It uses F-test to statistically test the equality of means and the relative variance between them.
Assumptions :
•Samples are random and independent.
•Homogeneity of sample variance.

NON PARAMETRIC
TEST
• Non-parametric test is a statistical analysis method that does not assume the population
data belongs to some prescribed distribution which is determined by some parameters.
• When the data does not meet the requirements to perform a parametric test, a non-
parametric test is used to analyze it.
 When the distribution is skewed, a non-parametric test is used.
 If the size of the data is too small then validating the distribution of the data
becomes difficult.
 If the data is nominal or ordinal, a non-parametric test is used.
• Types of non parametric test :
 Chi-square test
 Mann-Whitney U Test
 Wilcoxon Signed Rank Test
 Sign Test
 Kruskal Wallis Test

CHI-SQUARE TEST
A chi-squared test (symbolically represented as χ2) is basically a data analysis on the basis of
observations of a random set of variables.
It is a comparison of two statistical data sets.
When we consider, the null speculation is true, the sampling distribution of the test statistic is called
as chi-squared distribution.
Finding P-Value :
• P stands for probability here.
• To calculate the p-value, the chi-square test is used in statistics.
•P≤ 0.05; Hypothesis rejected
•P>.05; Hypothesis Accepted
Formula Used :
𝑥2
=
𝑜𝑖 − 𝜀𝑖
2
𝜀𝑖

MANN-WHITNEY U
TEST
• This non-parametric test is analogous to t-tests for independent samples. To conduct
such a test the distribution must contain ordinal data. It is also known as the
Wilcoxon rank sum test.
• Null Hypothesis: H0: The two populations under consideration must be equal.
• Test Statistic: U should be smaller of
OR
• where, R1R1 is the sum of ranks in group 1 and R2R2 is the sum of ranks in group
2.
• Decision Criteria: Reject the null hypothesis if U < critical value.
𝑈 = 𝑛1𝑛2 +
𝑛1 𝑛1 + 1
2
− 𝑅1 𝑈 = 𝑛1𝑛2 +
𝑛2 𝑛2 + 1
2
− 𝑅2

WILCOXON SIGNED RANK
TEST
• This is the non-parametric test whose counterpart is the
parametric paired t-test.
• It is used to compare two samples that contain ordinal data
and are dependent.
• The Wilcoxon signed rank test assumes that the data comes
from a symmetric distribution.
• Null Hypothesis: H0: The difference in the median is 0.
• Test Statistic: W. W is defined as the smaller of the sums of
the negative and positive ranks.
• Decision Criteria: Reject the null hypothesis if W < critical
value.

SIGN
TEST
• This non-parametric test is the parametric counterpart to
the paired samples t-test.
• The sign test is similar to the Wilcoxon sign test.
• Null Hypothesis: H0: The difference in the median is 0.
• Test Statistic: The smaller value among the number of
positive and negative signs.
• Decision Criteria: Reject the null hypothesis if the test
statistic < critical value.

KRUSKAL WALLIS
TEST
• The parametric one-way ANOVA test is analogous to the non-parametric
Kruskal Wallis test. It is used for comparing more than two groups of data
that are independent and ordinal.
• Null Hypothesis: H0H0: m population medians are equal
• Test Statistic:
𝐻 =
12
𝑁 𝑁 + 1
𝛴1
𝑚
𝑅𝑖
2
𝑛𝑖
− 3 𝑁 + 1
• where, N = total sample size, nj and Rj are the sample size and the sum
of ranks of the jth group
• Decision Criteria: Reject the null hypothesis if H > critical value

Hypothesis Testing.pptx

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