MEV 019: Unit 10 - Statistical Analysis-I
UNIT 10: STATISTICAL ANALYSIS – I
10.1
Introduction
Statistical analysis involves applying statistical
methods to draw conclusions from data. One of the most important applications
is hypothesis testing, which allows us to decide whether an assumption (hypothesis)
about a population parameter is supported by sample data.
This unit explains the concept of hypotheses, basic
terms in testing, types of errors, levels of significance, and large sample
tests for means and proportions.
10.2
Objectives
After completing this unit, you will be able to:
- Define
statistical hypotheses and differentiate between null and alternative
hypotheses.
- Understand
the meaning of Type I and Type II errors.
- Explain
the concept of critical regions, level of significance, and degrees of
freedom.
- Describe
the steps in hypothesis testing.
- Perform
one-tail and two-tail tests for large samples.
- Test
hypotheses regarding means and proportions.
10.3
Hypothesis
A statistical hypothesis is an assumption
about a population parameter that can be tested using sample data.
10.3.1 Null and Alternative Hypothesis
- Null
Hypothesis (H₀): States that there is no effect or no
difference. It represents the status quo.
Example: H₀: μ = 50 (Population mean equals 50) - Alternative
Hypothesis (H₁): States that there is an effect or
difference. It contradicts H₀.
Example: H₁: μ ≠ 50
10.3.2 Simple and Composite Hypothesis
- Simple
Hypothesis: Specifies the population distribution
completely. Example: μ = 100 and σ = 10.
- Composite
Hypothesis: Does not specify the distribution
completely. Example: μ > 100.
10.4 Some
Basic Concepts
10.4.1 Critical Regions (Region of Rejection)
The critical region is the set of values of the
test statistic that leads to rejection of H₀. The boundaries are determined by
the level of significance.
10.4.2 Type-I and Type-II Error
- Type I
Error (α): Rejecting H₀ when it is true.
- Type II
Error (β): Not rejecting H₀ when it is false.
10.4.3 Level of Significance
The probability of committing a Type I error,
usually set at 5% or 1%.
10.4.4 Degree of Freedom (df)
The number of independent values that can vary in
an analysis without breaking any constraints.
10.5 Testing
of Hypothesis
10.5.1 Procedure of Testing of Hypothesis
- State
H₀ and H₁ clearly.
- Choose the
level of significance (α).
- Select
the appropriate test statistic.
- Determine
the critical value(s) from statistical tables.
- Compute
the test statistic using sample data.
- Make a
decision: Compare computed value with critical value.
- State
the conclusion in the context of the problem.
10.5.2 One-tail and Two-tail Test
- One-tail
test: Critical region is in one tail of the
distribution (tests for greater than or less than).
- Two-tail
test: Critical region is in both tails (tests for
difference without direction).
10.6 Large
Sample Tests
Large sample tests are based on the normal
distribution and are generally applicable when n ≥ 30.
10.6.1 Test for the Significance of Population Mean
If σ is known:
Z=Xˉ−μσ/nZ = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}Z=σ/nXˉ−μ
If σ is unknown (but n is large), replace σ with
sample standard deviation s.
10.6.2 Test for Equality of Two Population Means
Z=Xˉ1−Xˉ2σ12n1+σ22n2Z = \frac{\bar{X}_1 -
\bar{X}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}Z=n1σ12+n2σ22Xˉ1−Xˉ2
If σ₁ and σ₂ are unknown, use sample variances.
10.6.3 Test for the Significance of Population
Proportion
Z=p−PP(1−P)nZ = \frac{p - P}{\sqrt{\frac{P(1-P)}{n}}}Z=nP(1−P)p−P
Where:
- p =
sample proportion
- P =
population proportion
10.6.4 Test for Equality of Two Population
Proportions
Z=p1−p2P(1−P)(1n1+1n2)Z = \frac{p_1 - p_2}{\sqrt{P(1-P)\left(\frac{1}{n_1}
+ \frac{1}{n_2}\right)}}Z=P(1−P)(n11+n21)p1−p2
Where:
P=x1+x2n1+n2P = \frac{x_1 + x_2}{n_1 + n_2}P=n1+n2x1+x2
10.7 Let Us
Sum Up
- A
hypothesis is an assumption about a population parameter.
- The
null hypothesis is tested against the alternative hypothesis.
- Type I
and Type II errors must be considered when making decisions.
- The
critical region and level of significance guide decision-making.
- Large
sample Z-tests are used for testing means and proportions when n ≥ 30.
10.8 Key
Words
- Null
Hypothesis (H₀) – Assumes no effect or difference.
- Alternative
Hypothesis (H₁) – Indicates a real effect or difference.
- Type I
Error (α) – Rejecting a true null hypothesis.
- Type II
Error (β) – Accepting a false null hypothesis.
- Level
of Significance – Probability of committing Type I error.
- Critical
Region – Range of values leading to rejection of H₀.
- Z-test – Statistical test for large samples.
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