259 lines
6.1 KiB
Markdown
259 lines
6.1 KiB
Markdown
# Quick Practice Drills — English
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## t-Tests, Confidence Intervals, Hypothesis Testing
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---
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## Drill 1: Simple t-Test
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**Given:**
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- Sample size: n = 38
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- Estimated coefficient: β̂ = 2.4
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- Standard error: SE = 0.9
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**Test:** H₀: β = 0 vs H₁: β ≠ 0 at α = 0.05
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**Your tasks:**
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1. Calculate the t-statistic
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2. Find degrees of freedom
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3. Find critical value (two-tailed, α = 0.05)
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4. Make your decision: Reject or fail to reject H₀?
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5. Calculate the p-value range using t-table
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<details>
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<summary>Answers</summary>
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1. t = (2.4 - 0) / 0.9 = 2.667
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2. df = 38 - 2 = 36 (simple regression)
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3. Critical value ≈ 2.028 (or use 2.042 for df=30, 2.021 for df=40)
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4. |2.667| > 2.028 → **Reject H₀**
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5. From t-table: 2.434 < 2.667 < 2.750 → **0.01 < p < 0.02**
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</details>
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---
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## Drill 2: One-Tailed Test
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**Given:**
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- n = 55
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- β̂ = -1.8, SE = 0.7
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**Test:** H₀: β ≥ 0 vs H₁: β < 0 at α = 0.01
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**Your tasks:**
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1. Calculate t-statistic
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2. Find critical value (one-tailed, α = 0.01)
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3. Decision?
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<details>
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<summary>Answers</summary>
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1. t = (-1.8 - 0) / 0.7 = -2.571 → |t| = 2.571
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2. df = 53, one-tailed critical at α=0.01: ≈ 2.404
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3. 2.571 > 2.404 → **Reject H₀** (evidence that β < 0)
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</details>
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---
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## Drill 3: 95% Confidence Interval
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**Given:**
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- n = 42
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- β̂ = 3.6, SE = 1.2
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**Your tasks:**
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1. Construct 95% confidence interval
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2. Interpret the interval
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3. Does this interval contain 2.0? What does that tell you?
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<details>
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<summary>Answers</summary>
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1. df = 40, t₀.₀₂₅ = 2.021
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Margin = 2.021 × 1.2 = 2.425
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CI: [3.6 - 2.425, 3.6 + 2.425] = **[1.175, 4.025]**
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2. We are 95% confident the true β lies between 1.175 and 4.025
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3. Yes, 2.0 is in the interval → We **cannot reject** H₀: β = 2.0 at α=0.05
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</details>
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---
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## Drill 4: Multiple Regression Test
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**Regression output:**
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| Variable | Coefficient | Std. Error |
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|----------|-------------|------------|
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| Intercept | 5.2 | 2.1 |
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| X₁ | **1.5** | **0.4** |
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| X₂ | -0.8 | 0.6 |
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| X₃ | 2.1 | 0.9 |
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n = 65
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**Your tasks:**
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1. Test each slope coefficient at α = 0.05
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2. Which variables are significant?
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3. Construct 90% CI for X₁
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<details>
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<summary>Answers</summary>
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1. df = 65 - 3 - 1 = 61, critical value ≈ 1.96 (or 2.000 for df=60)
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- X₁: t = 1.5/0.4 = 3.75 → **Significant** ✓
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- X₂: t = -0.8/0.6 = -1.33 → |1.33| < 2.0 → **Not significant** ✗
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- X₃: t = 2.1/0.9 = 2.33 → **Significant** ✓
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2. X₁ and X₃ are significant at 5% level
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3. 90% CI for X₁: t₀.₀₅,₆₁ ≈ 1.671
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Margin = 1.671 × 0.4 = 0.668
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CI: [1.5 - 0.668, 1.5 + 0.668] = **[0.832, 2.168]**
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</details>
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---
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## Drill 5: CI for Hypothesis Test
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**Given:**
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- 95% CI for β: [0.5, 3.2]
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**Test:** H₀: β = 4.0 vs H₁: β ≠ 4.0 at α = 0.05
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**Your task:** Use the CI method to test this hypothesis.
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<details>
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<summary>Answer</summary>
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4.0 lies **outside** the 95% CI [0.5, 3.2]
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→ **Reject H₀** at α = 0.05
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The hypothesized value 4.0 is not a plausible value for β.
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</details>
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---
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## Drill 6: Interpretation Check
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**Given:** β̂ = 2.3, p-value = 0.08
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**Which statements are correct?**
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- [ ] The effect is not statistically significant at 5% level
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- [ ] There is an 8% probability that β = 0
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- [ ] If H₀ were true, there's 8% chance of seeing this result
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- [ ] We are 92% confident there is an effect
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- [ ] At 10% level, the effect would be significant
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<details>
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<summary>Answers</summary>
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Correct: ✓✗✓✗✓
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- ✓ Not significant at 5% (0.08 > 0.05)
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- ✗ P-value is NOT probability H₀ is true
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- ✓ Correct interpretation of p-value
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- ✗ Confidence and significance are different concepts
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- ✓ Would be significant at 10% (0.08 < 0.10)
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</details>
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---
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## Drill 7: Full Problem — Coffee Shop Prices
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**Scenario:** A researcher studies coffee shop prices in 28 cities.
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**Model:** Priceᵢ = β₀ + β₁Rentᵢ + β₂Wageᵢ + uᵢ
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**Output:**
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| Variable | Coefficient | Std. Error |
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|----------|-------------|------------|
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| Intercept | 1.50 | 0.80 |
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| Rent (€100/m²) | **0.25** | **0.08** |
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| Wage (€/hour) | 0.15 | 0.12 |
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**Questions:**
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1. Test if rent affects price at 5% level
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2. Test if wage affects price at 5% level
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3. Construct 95% CI for rent coefficient
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4. A politician claims each €100 rent increase raises price by €0.40. Test this claim.
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<details>
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<summary>Answers</summary>
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1. Rent: t = 0.25/0.08 = 3.125, df = 25, critical ≈ 2.060
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3.125 > 2.060 → **Significant** ✓
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2. Wage: t = 0.15/0.12 = 1.25, |1.25| < 2.060 → **Not significant** ✗
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3. 95% CI for rent: t₀.₀₂₅,₂₅ = 2.060
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Margin = 2.060 × 0.08 = 0.165
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CI: [0.25 - 0.165, 0.25 + 0.165] = **[0.085, 0.415]**
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4. Politician claims β₁ = 0.40. Is 0.40 in the CI [0.085, 0.415]?
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Yes! (0.40 is inside the interval)
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→ **Cannot reject** the politician's claim at 5% level.
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</details>
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---
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## Drill 8: Quick Calculations
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**Calculate mentally or with scratch paper:**
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| β̂ | SE | n | k | t-stat | Significant at 5%? |
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|----|----|---|---|--------|-------------------|
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| 4.0 | 1.5 | 30 | 1 | ? | ? |
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| -2.5 | 1.0 | 50 | 2 | ? | ? |
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| 0.8 | 0.3 | 100 | 3 | ? | ? |
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| 1.2 | 0.9 | 25 | 1 | ? | ? |
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<details>
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<summary>Answers</summary>
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| β̂ | SE | n | k | t-stat | df | critical | Significant? |
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|----|----|---|---|--------|-----|----------|--------------|
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| 4.0 | 1.5 | 30 | 1 | 2.67 | 28 | 2.048 | **Yes** |
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| -2.5 | 1.0 | 50 | 2 | 2.50 | 47 | 2.012 | **Yes** |
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| 0.8 | 0.3 | 100 | 3 | 2.67 | 96 | 1.985 | **Yes** |
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| 1.2 | 0.9 | 25 | 1 | 1.33 | 23 | 2.069 | **No** |
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</details>
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---
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## Formula Sheet
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**t-statistic:**
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$$t = \frac{\hat{\beta} - \beta_0}{SE(\hat{\beta})}$$
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**Confidence Interval:**
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$$CI = \hat{\beta} \pm t_{\alpha/2, df} \times SE(\hat{\beta})$$
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**Degrees of freedom:**
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- Simple regression: df = n - 2
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- Multiple regression: df = n - k - 1 (where k = number of X variables)
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**Common critical values (two-tailed):**
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| df | α = 0.10 | α = 0.05 | α = 0.01 |
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|----|----------|----------|----------|
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| 25 | 1.708 | 2.060 | 2.787 |
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| 30 | 1.697 | 2.042 | 2.750 |
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| 40 | 1.684 | 2.021 | 2.704 |
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| 60 | 1.671 | 2.000 | 2.660 |
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| 120 | 1.658 | 1.980 | 2.617 |
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| ∞ | 1.645 | 1.960 | 2.576 |
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---
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*Practice these until you can do them in your sleep! 🎯* |