- Understand the concept of hypothesis testing.
- Differentiate between null and alternative hypotheses.
- Learn about significance levels, p-values, and decision rules.
- Identify and minimize Type I and Type II errors.
- Perform hypothesis tests and interpret results.
Definition of Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data.
A hypothesis is an assumption that we test using statistical techniques.
Types of Hypotheses and Errors
Hypothesis testing involves two competing hypotheses:
- Null Hypothesis (\( H_0 \)): Assumes no effect or no difference. It is what we test against.
- Alternative Hypothesis (\( H_a \)): Represents a significant effect or difference.
There are two types of errors in hypothesis testing:
| Type of Error | Definition |
|---|---|
| Type I Error (False Positive) | Rejecting \( H_0 \) when it is actually true. |
| Type II Error (False Negative) | Failing to reject \( H_0 \) when \( H_a \) is true. |
p-Value and Significance Level
The p-value is the probability of obtaining a result as extreme as the observed one, assuming \( H_0 \) is true.
- If \( p \)-value \( \leq \alpha \) (significance level), reject \( H_0 \).
- If \( p \)-value \( > \alpha \), fail to reject \( H_0 \).
Common Significance Levels:
| Significance Level (\( \alpha \)) | Confidence Level |
|---|---|
| 0.10 | 90% |
| 0.05 | 95% |
| 0.01 | 99% |
Interpretation of p-Values
The smaller the p-value, the stronger the evidence against \( H_0 \).
- p-value \( > 0.05 \): Weak evidence against \( H_0 \), fail to reject.
- p-value \( < 0.05 \): Moderate evidence against \( H_0 \), reject \( H_0 \).
- p-value \( < 0.01 \): Strong evidence against \( H_0 \), reject \( H_0 \).
Steps in Hypothesis Testing
The process of hypothesis testing follows these steps:
- State \( H_0 \) and \( H_a \).
- Choose a significance level \( \alpha \).
- Collect and summarize data.
- Compute the test statistic.
- Find the p-value and compare it with \( \alpha \).
- Make a decision: Reject \( H_0 \) if \( p \leq \alpha \), otherwise fail to reject.
Visualization of Hypothesis Testing
Examples
Example 1: A factory claims that its light bulbs last 1000 hours on average. A sample of 50 bulbs has a mean lifespan of 950 hours with a standard deviation of 80 hours. Test the claim at \( \alpha = 0.05 \).
Step 1: Hypotheses
- \( H_0: \mu = 1000 \)
- \( H_a: \mu < 1000 \) (one-tailed test)
Step 2: Compute Test Statistic
\[ z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} = \frac{950 - 1000}{80/\sqrt{50}} = -3.94 \]Step 3: Decision
- From the z-table, \( P(Z < -3.94) = 0.00004 \).
- Since \( p < 0.05 \), reject \( H_0 \).
Exercises
- Question 1: A coffee shop claims its coffee contains 250 mg of caffeine. A sample of 36 cups has a mean of 245 mg and a standard deviation of 15 mg. Test the claim at \( \alpha = 0.01 \).
- Question 2: A drug manufacturer claims their medication reduces blood pressure by 10 mmHg. A study finds a mean reduction of 9.2 mmHg with a standard deviation of 2.5 mmHg in 40 patients. Test at \( \alpha = 0.05 \).
- Question 3: A school claims students study an average of 3 hours per day. A sample of 50 students shows a mean of 2.8 hours and a standard deviation of 0.9 hours. Test at \( \alpha = 0.05 \).
- Answer 1: \( p = 0.043 \), fail to reject \( H_0 \).
- Answer 2: \( p = 0.067 \), fail to reject \( H_0 \).
- Answer 3: \( p = 0.015 \), reject \( H_0 \).