clawd/econometrics-quick-drills.md

Quick Practice Drills — English

t-Tests, Confidence Intervals, Hypothesis Testing

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Drill 1: Simple t-Test

Given:

• Sample size: n = 38
• Estimated coefficient: β̂ = 2.4
• Standard error: SE = 0.9

Test: H₀: β = 0 vs H₁: β ≠ 0 at α = 0.05

Your tasks:

1. Calculate the t-statistic
2. Find degrees of freedom
3. Find critical value (two-tailed, α = 0.05)
4. Make your decision: Reject or fail to reject H₀?
5. Calculate the p-value range using t-table

Answers

1. t = (2.4 - 0) / 0.9 = 2.667
2. df = 38 - 2 = 36 (simple regression)
3. Critical value ≈ 2.028 (or use 2.042 for df=30, 2.021 for df=40)
4. |2.667| > 2.028 → Reject H₀
5. From t-table: 2.434 < 2.667 < 2.750 → 0.01 < p < 0.02

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Drill 2: One-Tailed Test

Given:

• n = 55
• β̂ = -1.8, SE = 0.7

Test: H₀: β ≥ 0 vs H₁: β < 0 at α = 0.01

Your tasks:

1. Calculate t-statistic
2. Find critical value (one-tailed, α = 0.01)
3. Decision?

Answers

1. t = (-1.8 - 0) / 0.7 = -2.571 → |t| = 2.571
2. df = 53, one-tailed critical at α=0.01: ≈ 2.404
3. 2.571 > 2.404 → Reject H₀ (evidence that β < 0)

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Drill 3: 95% Confidence Interval

Given:

• n = 42
• β̂ = 3.6, SE = 1.2

Your tasks:

1. Construct 95% confidence interval
2. Interpret the interval
3. Does this interval contain 2.0? What does that tell you?

Answers

1. df = 40, t₀.₀₂₅ = 2.021 Margin = 2.021 × 1.2 = 2.425 CI: [3.6 - 2.425, 3.6 + 2.425] = [1.175, 4.025]

2. We are 95% confident the true β lies between 1.175 and 4.025

3. Yes, 2.0 is in the interval → We cannot reject H₀: β = 2.0 at α=0.05

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Drill 4: Multiple Regression Test

Regression output:

Variable	Coefficient	Std. Error
Intercept	5.2	2.1
X₁	1.5	0.4
X₂	-0.8	0.6
X₃	2.1	0.9

n = 65

Your tasks:

1. Test each slope coefficient at α = 0.05
2. Which variables are significant?
3. Construct 90% CI for X₁

Answers

1. df = 65 - 3 - 1 = 61, critical value ≈ 1.96 (or 2.000 for df=60)

   • X₁: t = 1.5/0.4 = 3.75 → Significant ✓
   • X₂: t = -0.8/0.6 = -1.33 → |1.33| < 2.0 → Not significant ✗
   • X₃: t = 2.1/0.9 = 2.33 → Significant ✓

2. X₁ and X₃ are significant at 5% level

3. 90% CI for X₁: t₀.₀₅,₆₁ ≈ 1.671 Margin = 1.671 × 0.4 = 0.668 CI: [1.5 - 0.668, 1.5 + 0.668] = [0.832, 2.168]

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Drill 5: CI for Hypothesis Test

Given:

• 95% CI for β: [0.5, 3.2]

Test: H₀: β = 4.0 vs H₁: β ≠ 4.0 at α = 0.05

Your task: Use the CI method to test this hypothesis.

Answer

4.0 lies outside the 95% CI [0.5, 3.2]

→ Reject H₀ at α = 0.05

The hypothesized value 4.0 is not a plausible value for β.

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Drill 6: Interpretation Check

Given: β̂ = 2.3, p-value = 0.08

Which statements are correct?

• The effect is not statistically significant at 5% level
• There is an 8% probability that β = 0
• If H₀ were true, there's 8% chance of seeing this result
• We are 92% confident there is an effect
• At 10% level, the effect would be significant

Answers

Correct: ✓✗✓✗✓

• ✓ Not significant at 5% (0.08 > 0.05)
• ✗ P-value is NOT probability H₀ is true
• ✓ Correct interpretation of p-value
• ✗ Confidence and significance are different concepts
• ✓ Would be significant at 10% (0.08 < 0.10)

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Drill 7: Full Problem — Coffee Shop Prices

Scenario: A researcher studies coffee shop prices in 28 cities.

Model: Priceᵢ = β₀ + β₁Rentᵢ + β₂Wageᵢ + uᵢ

Output:

Variable	Coefficient	Std. Error
Intercept	1.50	0.80
Rent (€100/m²)	0.25	0.08
Wage (€/hour)	0.15	0.12

Questions:

1. Test if rent affects price at 5% level
2. Test if wage affects price at 5% level
3. Construct 95% CI for rent coefficient
4. A politician claims each €100 rent increase raises price by €0.40. Test this claim.

Answers

1. Rent: t = 0.25/0.08 = 3.125, df = 25, critical ≈ 2.060 3.125 > 2.060 → Significant ✓

2. Wage: t = 0.15/0.12 = 1.25, |1.25| < 2.060 → Not significant ✗

3. 95% CI for rent: t₀.₀₂₅,₂₅ = 2.060 Margin = 2.060 × 0.08 = 0.165 CI: [0.25 - 0.165, 0.25 + 0.165] = [0.085, 0.415]

4. Politician claims β₁ = 0.40. Is 0.40 in the CI [0.085, 0.415]? Yes! (0.40 is inside the interval) → Cannot reject the politician's claim at 5% level.

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Drill 8: Quick Calculations

Calculate mentally or with scratch paper:

β̂	SE	n	k	t-stat	Significant at 5%?
4.0	1.5	30	1	?	?
-2.5	1.0	50	2	?	?
0.8	0.3	100	3	?	?
1.2	0.9	25	1	?	?

Answers

β̂	SE	n	k	t-stat	df	critical	Significant?
4.0	1.5	30	1	2.67	28	2.048	Yes
-2.5	1.0	50	2	2.50	47	2.012	Yes
0.8	0.3	100	3	2.67	96	1.985	Yes
1.2	0.9	25	1	1.33	23	2.069	No

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Formula Sheet

t-statistic:

t = \frac{\hat{\beta} - \beta_0}{SE(\hat{\beta})}

Confidence Interval:

CI = \hat{\beta} \pm t_{\alpha/2, df} \times SE(\hat{\beta})

Degrees of freedom:

• Simple regression: df = n - 2
• Multiple regression: df = n - k - 1 (where k = number of X variables)

Common critical values (two-tailed):

df	α = 0.10	α = 0.05	α = 0.01
25	1.708	2.060	2.787
30	1.697	2.042	2.750
40	1.684	2.021	2.704
60	1.671	2.000	2.660
120	1.658	1.980	2.617
∞	1.645	1.960	2.576

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Practice these until you can do them in your sleep! 🎯