Statistical Estimation and Hypothesis Testing

Descriptive Statistics Review

Measures of Central Tendency

Measures of Dispersion

Note: For sample statistics, divide by (n-1), not n. This is called Bessel's correction and produces an unbiased estimate of population variance.

Point Estimation

A point estimate is a single value used to estimate a population parameter:

Confidence Intervals

A confidence interval provides a range of plausible values for a population parameter.

For Population Mean (σ known):

$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

For Population Mean (σ unknown, use t):

$\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}$

Common z-values:

Example: A sample of 25 concrete cylinders has x̄ = 4,500 psi and s = 300 psi. Find the 95% confidence interval for the population mean.

With n = 25, df = 24, t₀.₀₂₅ ≈ 2.064: CI = 4,500 ± 2.064 × (300/√25) = 4,500 ± 123.8 = (4,376.2, 4,623.8) psi

Hypothesis Testing

Steps:

1. State hypotheses: H₀ (null) and H₁ (alternative)
2. Choose significance level α (commonly 0.05)
3. Calculate test statistic
4. Compare to critical value or compute p-value
5. Make decision: Reject H₀ if |test statistic| > critical value

Test Statistic for Mean:

$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \quad \text{or} \quad t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

Types of Errors

• Type I error (α): False positive — rejecting a true null hypothesis
• Type II error (β): False negative — failing to reject a false null hypothesis
• Power (1-β): Probability of correctly rejecting a false null hypothesis

Regression and Correlation

Linear Regression

The least-squares line: ŷ = b₀ + b₁x

$b_1 = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2}$

$b_0 = \bar{y} - b_1 \bar{x}$

Coefficient of Determination (R²)

• R² ranges from 0 to 1
• R² = 1 means perfect fit (all variation explained by the model)
• R² = 0 means the model explains none of the variation
• R² = 0.85 means 85% of the variation in y is explained by x

Correlation Coefficient (r)

• r ranges from -1 to +1
• r = +1: perfect positive linear relationship
• r = -1: perfect negative linear relationship
• r = 0: no linear relationship
• r = ±√(R²) — sign indicates direction