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Introduction to StatisticsTopic 23 of 24

Overview of Hypothesis Testing

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1. What is Hypothesis Testing?

Definition: Hypothesis testing is a statistical method used to make decisions or inferences about a population parameter based on sample data.

It helps us decide whether to accept or reject a claim (hypothesis) about a population.
Provides a formal framework for testing assumptions using data.

2. Key Concepts

Term	Meaning
Population Parameter	A numerical characteristic of a population (e.g., mean $\mu$ , proportion $p$ )
Sample Statistic	A value computed from a sample (e.g., sample mean $\bar{X}$ )
Null Hypothesis ( $H_0$ )	The statement of no effect or no difference. It is assumed true unless evidence suggests otherwise.
Alternative Hypothesis ( $H_1$ or $H_a$ )	The statement that contradicts $H_0$ , representing the effect or difference we want to detect.
Significance Level ( $\alpha$ )	Probability of rejecting $H_0$ when it is true (Type I error). Common values: 0.05, 0.01
Test Statistic	A standardized value calculated from sample data used to make a decision (e.g., $z$ , $t$ , $\chi^2$ ).
P-value	Probability of obtaining a result as extreme as observed if $H_0$ is true.
Critical Region / Rejection Region	The set of values of the test statistic for which $H_0$ is rejected.

3. Steps in Hypothesis Testing

Formulate Hypotheses
- Null hypothesis ( $H_0$ ): "No effect"
- Alternative hypothesis ( $H_1$ or $H_a$ ): "Effect exists"
Choose Significance Level ( $\alpha$ )
- Common: 0.05 (5%), 0.01 (1%)
Select the Appropriate Test
- Based on type of data, sample size, and population distribution
- Examples: z-test, t-test, chi-square test, F-test
Compute Test Statistic
- Compare sample statistic to hypothesized population parameter
Determine Critical Value or P-value
- Using statistical tables or software
Make a Decision
- If test statistic falls in rejection region or p-value < $\alpha$ : Reject $H_0$
- Otherwise: Fail to reject $H_0$
Draw a Conclusion
- State result in the context of the problem

4. Types of Hypothesis Tests

One-Tailed Test
- Tests directional hypothesis (e.g., mean > 50)
Two-Tailed Test
- Tests non-directional hypothesis (e.g., mean ≠ 50)

5. Types of Errors in Hypothesis Testing

Error Type	Description	Probability
Type I Error	Reject $H_0$ when it is true	$\alpha$
Type II Error	Fail to reject $H_0$ when it is false	$\beta$

Note:

Smaller $\alpha$ → less chance of Type I error but may increase Type II error.

6. Example

Problem: Test whether the mean weight of a population is 70 kg.

$H_0: \mu = 70$
$H_1: \mu \ne 70$ (two-tailed test)
Sample mean = 72, sample size = 25, population $\sigma = 5$

Test statistic (z):

z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} = \frac{72-70}{5/\sqrt{25}} = \frac{2}{1} = 2

At $\alpha = 0.05$ , two-tailed critical z-values = ±1.96
Since $2 > 1.96$ , reject $H_0$ → mean weight is significantly different from 70.

7. Key Points to Remember

Hypothesis testing does not prove a hypothesis; it provides evidence to support or reject it.
Sample data is used to make inference about the population.
Choosing the right test and correct significance level is crucial.
Understanding Type I and Type II errors helps in interpreting results.

Previous topic 22

Sampling Distributions for Difference of Means and Difference of Proportions

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Overview of Regression Analysis

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2. Key Concepts

Term

Meaning

Population Parameter

A numerical characteristic of a population (e.g., mean

\mu

, proportion

p

)

Sample Statistic

A value computed from a sample (e.g., sample mean

\bar{X}

)

Null Hypothesis ( $H_0$ )

The statement of no effect or no difference. It is assumed true unless evidence suggests otherwise.

Alternative Hypothesis ( $H_1$ or $H_a$ )

The statement that contradicts $H_0$ , representing the effect or difference we want to detect.

Significance Level ( $\alpha$ )

Probability of rejecting $H_0$ when it is true (Type I error). Common values: 0.05, 0.01

Test Statistic

A standardized value calculated from sample data used to make a decision (e.g.,

z

t

\chi^2

P-value

Probability of obtaining a result as extreme as observed if $H_0$ is true.

Critical Region / Rejection Region

The set of values of the test statistic for which

H_0

is rejected.

3. Steps in Hypothesis Testing

Formulate Hypotheses

Null hypothesis ( $H_0$ ): "No effect"
Alternative hypothesis ( $H_1$ or $H_a$ ): "Effect exists"

Choose Significance Level ( $\alpha$ )

Common: 0.05 (5%), 0.01 (1%)

Select the Appropriate Test

Based on type of data, sample size, and population distribution
Examples: z-test, t-test, chi-square test, F-test

Compute Test Statistic

Compare sample statistic to hypothesized population parameter

Determine Critical Value or P-value

Using statistical tables or software

Make a Decision

If test statistic falls in rejection region or p-value < $\alpha$ : Reject $H_0$
Otherwise: Fail to reject $H_0$

Draw a Conclusion

State result in the context of the problem

Error Type

Description

Probability

Type I Error

Reject

H_0

when it is true

\alpha

Type II Error

Fail to reject

H_0

when it is false

\beta

6. Example

Problem: Test whether the mean weight of a population is 70 kg.

H_0: \mu = 70

H_1: \mu \ne 70

(two-tailed test)

Sample mean = 72, sample size = 25, population

\sigma = 5

Test statistic (z):

z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} = \frac{72-70}{5/\sqrt{25}} = \frac{2}{1} = 2

\alpha = 0.05

, two-tailed critical z-values = ±1.96

Since

2 > 1.96

, reject $H_0$ → mean weight is significantly different from 70.