Significance Test for a Population Mean

This video explains how to conduct a statistical significance test for a population mean (specifically using a T-test). It uses a practical example involving the underwater time of professional divers in the US.

You’ll learn how to set up hypotheses, calculate the T-score (since the population standard deviation is unknown), and compare it against a critical value to make a decision. The video also covers one-tailed vs. two-tailed tests and the impact of significance levels ($\alpha$).

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🧠 MAIN POINTS

• Hypothesis Testing for Mean: Focuses on the average value ($\mu$) of a population.
• T vs. Z: If the population standard deviation ($\sigma$) is unknown, use the sample standard deviation ($S$) → use the T-distribution.
• T-Score Formula: \(T = \frac{\bar{x} - \mu}{\frac{S}{\sqrt{n}}}\)
• Critical Value Decision: Compare the T-score to the critical value based on degrees of freedom and your $\alpha$ level.

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🔬 CASE STUDY: Oxygen Endurance of Divers

Scenario:

Do US divers stay underwater more than 60 minutes?

• Null Hypothesis ($H_0$): $\mu = 60$
• Alternative Hypothesis ($H_a$): $\mu > 60$ (One-tailed)

Sample Data:

• Sample Size ($n$): 100
• Sample Mean ($\bar{x}$): 62
• Sample Std Dev ($S$): 5

Step 1: Standard Error

\(SE = \frac{5}{\sqrt{100}} = 0.5\)

Step 2: T-Score

\(T = \frac{62 - 60}{0.5} = 4\)

Step 3: Critical Value

• Degrees of Freedom: $n - 1 = 99$
• Critical T-value (( \alpha = 0.05 ), one-tailed): ≈ 1.67

✅ Decision:

Since $4 > 1.67$ → Reject $H_0$
✔ We have strong evidence the mean is greater than 60.

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🔁 Two-Tailed Check (Stricter Test)

Test for $\mu \neq 60$ with $\alpha = 0.01$ (two-tailed):
Critical values: ±2.66
Result: $4 > 2.66$ → Still reject $H_0$
✔ Result is highly significant even with stricter conditions.

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🧪 PRACTICAL PLAN: How to Run a T-Test

Phase 1: Setup

• Define $H_0$: e.g., “Average time = 60”
• Define $H_a$: e.g., “Average time > 60”
• Assumptions: If $n < 30$, data should be normally distributed

Phase 2: Calculation

• \(SE = \frac{S}{\sqrt{n}}\)
• \(T = \frac{\bar{x} - \mu}{SE}\)

Phase 3: Decision

• Find critical value from T-table (based on $n - 1$ and $\alpha$)
• Compare T-score with critical value

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A step-by-step visual guide for conducting a T-test for population means. It covers setting up hypotheses, computing the T-score, finding the critical value based on degrees of freedom, and deciding whether to reject the null hypothesis.

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✅ Best Practices for T-Test for Means

• 🧪 Use a T-test when population standard deviation (σ) is unknown
• 📈 Report Degrees of Freedom (n − 1) when using t-distribution
• 📏 Check normality for small samples (n < 30) or rely on CLT for large samples
• 📊 Use one-tailed tests only with strong theoretical reasoning
• 📝 Clearly state your Null and Alternative Hypotheses in context

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⚠ Common Pitfalls

• 🚫 Using Z-test when σ is unknown — use t-distribution instead
• 🔁 Incorrect degrees of freedom can affect critical values
• 🔍 Forgetting to check assumptions — normality or sample size adequacy
• 🤷‍♀️ Using a one-tailed test without justification
• 📉 Misinterpreting P-values — they don’t measure probability of hypotheses

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🧠 Level-Up: Effect Size Matters

Even if your result is statistically significant, ask: Is it practically significant? Calculate Cohen's d to measure the effect size:

\[ d = \frac{\bar{x} - \mu_0}{s} \]

• Small effect: d = 0.2
• Medium effect: d = 0.5
• Large effect: d = 0.8+

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🧬 Why It Matters in Machine Learning

• 🔎 Model Validation: T-tests help confirm if model performance metrics differ significantly between versions
• 📊 Feature Impact: Test if the average value of a feature differs across classes (e.g., fraud vs non-fraud)
• ⚖ Baseline Comparison: Validate uplift over baselines using sample means

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📌 Try It Yourself: T-Test Quiz

Q1: When should you use the t-distribution instead of z-distribution?

💡 Show Answer

When the population standard deviation is unknown.

Q2: What’s the T-score formula?

💡 Show Answer

\[ T = \frac{\bar{x} - \mu_0}{SE} \] Where SE is the standard error, \( \frac{s}{\sqrt{n}} \)

Q3: Why is it important to report degrees of freedom?

💡 Show Answer

Because it determines the critical value in the t-distribution.

Q4: What does a small P-value mean?

💡 Show Answer

It means the observed result is unlikely under the Null Hypothesis — potential evidence against it.

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🧾 Summary

T-tests for means help determine whether your sample’s average truly differs from a known or hypothesized value. Always check assumptions, use the right distribution (t, not z), and understand both statistical and practical significance. A strong conclusion needs both correct math and context-aware interpretation.

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🎥 Hoda Osama AI

Learn statistics and machine learning concepts step by step with visuals and real examples.

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