Hypothesis testing helps us make decisions about populations based on sample data using statistical evidence.
1. State hypotheses (H₀ and H₁)
2. Choose significance level (α)
3. Calculate test statistic
4. Find p-value
5. Make decision
Type I Error: Reject true H₀
Type II Error: Accept false H₀
α (alpha): P(Type I Error)
Power: 1 - P(Type II Error)
Scenario: A factory claims their light bulbs last 1000 hours on average. A sample of 25 bulbs has a mean of 950 hours with standard deviation 100 hours. Test at α = 0.05.
Step 1: H₀: μ = 1000, H₁: μ ≠ 1000
Step 2: α = 0.05 (two-tailed)
Step 3: t = (x̄ - μ) / (s/√n)
t = (950 - 1000) / (100/√25) = -50/20 = -2.5
Step 4: With df = 24, |t| = 2.5
Step 5: Critical value ≈ ±2.064
Since |2.5| > 2.064, we reject H₀
Interpretation: There is sufficient evidence that the mean lifetime is not 1000 hours.
Practical meaning: The bulbs may not last as long as claimed.
Range: -1 ≤ r ≤ 1
r = 1: Perfect positive correlation
r = 0: No linear correlation
r = -1: Perfect negative correlation
|r| > 0.7: Strong correlation
Equation: y = mx + b
Slope (m): Change in y per unit x
y-intercept (b): Value when x = 0
R²: Proportion of variance explained
Question 1: If the p-value = 0.03 and α = 0.05, what is your decision about H₀?
Question 2: A correlation coefficient r = -0.85 indicates what type of relationship?
Question 3: In a two-tailed test with α = 0.01, what would be the critical z-values?