Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It evaluates whether there is enough evidence to support or reject a specific claim about a population parameter.

Key Concepts in Hypothesis Testing

Null Hypothesis ($H_0$)
The default assumption that there is no effect or difference.
- Example: $H_0: \mu = 50$ (The population mean is 50).
Alternative Hypothesis ($H_a$)
The statement being tested, representing a potential effect or difference.
- Example: $H_a: \mu \neq 50$ (The population mean is not 50).
Significance Level ($\alpha$)
The threshold for rejecting the null hypothesis, typically set at 0.05 (5%).
Test Statistic
A value calculated from the sample data used to compare with the critical value or p-value.
P-Value
The probability of observing a test statistic as extreme as, or more extreme than, the one observed, assuming $H_0$ is true.
Decision Rule
- Reject $H_0$ if $p$-value $\leq \alpha$.
- Fail to reject $H_0$ if $p$-value $> \alpha$.

Steps in Hypothesis Testing

State the Hypotheses:
- Null hypothesis ($H_0$)
- Alternative hypothesis ($H_a$)
Set the Significance Level ($\alpha$):
Common values are 0.01, 0.05, or 0.10.
Choose the Test and Calculate the Test Statistic:
Select the appropriate test (e.g., $t$-test, $z$-test) based on data type and sample size.
Find the P-Value or Critical Value:
Compare the test statistic to the critical value or use the $p$-value approach.
Make a Decision:
- Reject $H_0$ if there is sufficient evidence.
- Fail to reject $H_0$ if there is insufficient evidence.
Draw a Conclusion:
State the results in the context of the problem.

Types of Hypothesis Tests

1. One-Tailed Test

Tests if a parameter is greater than or less than a specific value.
Example: $H_0: \mu \leq 50, \, H_a: \mu > 50$

2. Two-Tailed Test

Tests if a parameter is not equal to a specific value.
Example: $H_0: \mu = 50, \, H_a: \mu \neq 50$

Common Tests in Hypothesis Testing

Z-Test
- Used for large sample sizes ($n > 30$) or when population variance is known.
- Example: Testing if the mean height of students differs from 170 cm.
T-Test
- Used for small sample sizes ($n \leq 30$) or when population variance is unknown.
- Types: One-sample $t$-test, two-sample $t$-test, paired $t$-test.
Chi-Square Test
- Used for categorical data to test relationships or goodness of fit.
- Example: Testing if the observed frequencies of colors in candy bags match the expected proportions.
ANOVA (Analysis of Variance)
- Used to compare means across three or more groups.
- Example: Testing if the average test scores differ across schools.

Example: One-Sample T-Test

Problem:

A company claims the average lifetime of its batteries is 300 hours. A random sample of 25 batteries has a mean lifetime of 290 hours with a standard deviation of 20 hours. Is there evidence at the 0.05 significance level to reject the company’s claim?

Solution:

State the Hypotheses:
$H_0: \mu = 300$, $H_a: \mu \neq 300$
Set the Significance Level:
$\alpha = 0.05$
Calculate the Test Statistic:
\[t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} = \frac{290 - 300}{\frac{20}{\sqrt{25}}} = \frac{-10}{4} = -2.5\]
Find the Critical Value or P-Value:
From $t$-distribution tables, critical $t$-value at $\alpha = 0.05$ (two-tailed, $df = 24$) is approximately $\pm 2.064$.
Decision:
Since $-2.5 < -2.064$, reject $H_0$.
Conclusion:
There is sufficient evidence to reject the claim that the average battery lifetime is 300 hours.

Applications of Hypothesis Testing

Healthcare: Testing the effectiveness of new treatments.
Business: Comparing customer satisfaction between two products.
Education: Evaluating the impact of new teaching methods.

Conclusion

Hypothesis testing is a powerful tool for making data-driven decisions. By systematically evaluating claims and interpreting results, it enables researchers and decision-makers to draw reliable conclusions.

Next Steps: Simple Linear Regression