Step 1: Set up hypotheses H₀: β₁ = 0 (no relationship) Hₐ: β₁ ≠ 0 (relationship exists)

Two-tailed test, α = 0.05

Step 2: Check conditions LINEAR: Assume scatterplot is linear ✓ INDEPENDENT: Assume random sample, n < 10% population ✓ NORMAL: Residuals approximately normal ✓ EQUAL VARIANCE: Residual plot shows constant spread ✓ RANDOM: Random sample ✓

(LINE conditions for regression inference)

Step 3: Calculate test statistic df = n - 2 = 25 - 2 = 23

t = (b₁ - 0)/SE t = 3.5/1.2 t ≈ 2.917

Step 4: Find p-value From t-table with df = 23, two-tailed: t = 2.917 is between t = 2.807 (p = 0.01) and t = 3.767 (p = 0.001)

So: 0.001 < p-value < 0.01

More precisely: p-value ≈ 0.0077

Step 5: Make decision p-value (0.0077) < α (0.05) REJECT H₀

Step 6: Conclusion in context "There is significant evidence (p = 0.008) that a linear relationship exists between x and y. The slope is significantly different from zero."

Answer: t = 2.92, p-value ≈ 0.008. Reject H₀. Significant evidence of linear relationship.

The LINE conditions for regression inference:

L - LINEAR Relationship between x and y is linear Check: Scatterplot should show linear pattern Residual plot should show no curve

I - INDEPENDENT
Observations are independent Check: Random sampling n < 10% of population (if sampling without replacement) No time series or repeated measures

N - NORMAL Residuals are approximately normally distributed Check: Histogram or normal probability plot of residuals Not critical if n is large (n ≥ 30) Just need no strong skewness or outliers

E - EQUAL VARIANCE (also called homoscedasticity) Variability of y is constant for all x Check: Residual plot shows roughly equal vertical spread No fan shape or other pattern in spread

Why these matter:

• LINEAR: For model to be appropriate
• INDEPENDENT: For formulas to be valid
• NORMAL: For t-distribution to apply (especially small samples)
• EQUAL VARIANCE: For standard errors to be correct

If violations:

• Not linear → transform or use nonlinear model
• Not independent → use different methods (time series, etc.)
• Not normal → okay if n ≥ 30; otherwise transform
• Not equal variance → transform or use weighted regression

Answer: LINE = Linear relationship, Independent observations, Normal residuals, Equal variance. Check using scatterplot, residual plot, and normal probability plot.

Step 1: Identify the test Testing: H₀: β₁ = 0 (no relationship) Against: Hₐ: β₁ ≠ 0 (relationship exists)

Given: p-value = 0.006

Step 2: What p-value means statistically The probability of observing a slope as extreme as 2.4 (or more extreme) IF the true slope is actually 0.

Step 3: Interpret in context "If there were truly no linear relationship between x and y (β₁ = 0), the probability of obtaining a sample slope of 2.4 or more extreme (in either direction) is 0.006, or 0.6%."

Step 4: Practical interpretation This is very unlikely (less than 1% chance)!

Therefore: Strong evidence AGAINST H₀ The relationship is statistically significant.

Step 5: Decision at α = 0.05 Since p-value (0.006) < α (0.05): REJECT H₀

Conclusion: "There is strong evidence of a significant linear relationship. The slope is significantly different from zero (p = 0.006)."

Step 6: What this does NOT mean ✗ Does not mean slope is definitely 2.4 ✗ Does not mean x causes y ✗ Does not mean model fits well (could still have problems) ✓ Only means: slope significantly different from zero

Answer: If true slope were 0, probability of getting b₁ = 2.4 or more extreme is only 0.006. This provides strong evidence the slope is not zero - there is a significant linear relationship.

Step 1: Compare to one-sample t-test One-sample t-test: df = n - 1

• Estimate 1 parameter: μ
• Lose 1 df

Regression: df = n - 2

• Estimate 2 parameters: β₀ AND β₁
• Lose 2 df

Step 2: What we're estimating In regression, we estimate:

1. Intercept (β₀)
2. Slope (β₁)

Both use up degrees of freedom!

Step 3: Degrees of freedom explained Start with n observations

• Use one to estimate β₀ (intercept)
• Use one to estimate β₁ (slope)
• Left with n - 2 for error estimation

df = n - 2

Step 4: Why it matters Smaller df → wider t* critical values → wider CIs

Example: n = 10, 95% confidence

• One-sample (df = 9): t* = 2.262
• Regression (df = 8): t* = 2.306

Regression CI slightly wider (more uncertainty).

Step 5: As n increases For large n, the difference is minimal:

• df = 100 vs 98 → nearly same t*
• Both approach z* = 1.96

Step 6: General pattern Degrees of freedom = n - (number of parameters estimated)

• Mean only: n - 1
• Regression: n - 2
• Multiple regression with k predictors: n - k - 1

Answer: We estimate TWO parameters (β₀ and β₁), so we lose 2 degrees of freedom, giving df = n - 2. This accounts for the extra uncertainty from estimating both intercept and slope.