Statistical Inference & Hypothesis Testing: Tests, Confidence, and Decision-Making

TAKEAWAYS:

• Statistical inference lets us estimate population parameters from sample data (e.g., average wage, market share).
• Hypothesis testing uses p-values and significance levels to decide whether observed data supports a claim (e.g., "Does eSewa’s transaction volume increase with promotions?").
• Confidence intervals quantify uncertainty in estimates (e.g., "Ncell’s 95% CI for call drop rate is 0.02–0.05%").
• Z-tests and t-tests compare sample means to known values or between groups (e.g., Daraz’s order delivery times before/after warehouse upgrades).
• Chi-square tests check if categorical data fits expected distributions (e.g., "Does Pathao’s surge pricing correlate with traffic density?").
• Type I/II errors have real costs: false alarms (e.g., recalling safe products) vs. missed defects (e.g., approving unsafe loans).

1. Statistical Inference: From Samples to Populations

Statistical inference is the process of making conclusions about a population based on data from a sample. Since we rarely survey every worker, customer, or stock (population), we use samples to generalize.

Key Concepts

• Population: The entire group of interest (e.g., all Ncell subscribers in Kathmandu).
• Sample: A subset of the population (e.g., 500 randomly selected Ncell users).
• Parameter: A fixed value describing the population (e.g., mean call duration).
• Statistic: A calculated value from the sample (e.g., sample mean call duration).

Types of Inference

1. Estimation: Guessing a population parameter (e.g., "What’s the average loan default rate in Nepal?").

   • Point estimate: Single value (e.g., sample mean = 3.2%).
   • Interval estimate: Range with confidence (e.g., 95% CI: 2.8%–3.6%).

2. Hypothesis Testing: Testing claims about the population (e.g., "Does Khalti’s transaction fee decrease after a rate cut?").

FIGURE 1: Population vs. Sample

figure:

graph TD
    A["Population: All Ncell users in Kathmandu (1M)"] -->|"Sample"| B["Sample: 500 users surveyed"]
    B --> C["Statistic: Sample mean call duration = 4.1 min"]
    C --> D["Parameter: True population mean (unknown)"]

Confidence Intervals

A confidence interval (CI) gives a range of values likely to contain the population parameter, with a given confidence level (e.g., 95%).

Formula for Mean (Large Sample, σ known): $$CI=\bar{x}\pm z_{\alpha /2}\cdot \frac{\sigma }{\sqrt{n}}$$ Where:

• $\bar{x}$ = sample mean
• $z_{\alpha /2}$ = critical z-value (e.g., 1.96 for 95% CI)
• $\sigma$ = population standard deviation
• $n$ = sample size

Worked Example: Daraz’s Order Delivery Time Daraz claims its average order delivery time is ≤48 hours. A sample of 100 orders has a mean of 50 hours and σ=12 hours. Test at 95% CI.

1. Calculate CI: $$CI=50\pm 1.96\cdot \frac{12}{\sqrt{100}}=50\pm 2.352⟹[47.648,52.352]$$ Since 48 hours is within the CI, we cannot reject Daraz’s claim at 95% confidence.

2. Interpretation:

   • 95% CI: We’re 95% sure the true mean delivery time is between 47.65 and 52.35 hours.
   • Business implication: Daraz’s claim is plausible, but the upper bound suggests delays exceed 48 hours.

FIGURE 2: Confidence Interval for Daraz’s Delivery Time

figure:

graph LR
    A["Sample Mean: 50 hours"] --> B["± Margin of Error: ±2.352"]
    B --> C["95% CI: [47.65, 52.35] hours"]
    C --> D["Claim: ≤48 hours"]
    D -->|"Within CI"| E["Fail to reject claim"]

Advantages/Disadvantages

2. Hypothesis Testing: Testing Claims

Hypothesis testing answers: "Is there enough evidence to support a claim?" Steps:

1. State hypotheses:
   • $H_0$: Null hypothesis (default claim, e.g., "No effect").
   • $H_a$: Alternative hypothesis (what we test, e.g., "There is an effect").
2. Choose significance level ($\alpha$, e.g., 0.05).
3. Calculate test statistic (z, t, chi-square).
4. Compare to critical value or p-value.
5. Make a decision: Reject $H_0$ or fail to reject.

FIGURE 3: Hypothesis Testing Workflow

figure:

graph TD
    A["State Hypotheses"] --> B["Choose α (e.g., 0.05)"]
    B --> C["Calculate Test Statistic"]
    C --> D["Compare to Critical Value or p-value"]
    D --> E["Decision: Reject/Fail to Reject H₀"]

Types of Tests

Worked Example: eSewa Transaction Volume

eSewa claims its daily transaction volume increased after a new app update. Before: mean = 50,000 transactions/day ($n=30$, σ=5,000). After: mean = 52,000 ($n=30$, σ=5,000). Test at α=0.05.

1. State hypotheses:

   • $H_0$: $\mu =50,000$ (no increase).
   • $H_a$: $\mu >50,000$ (increase).

2. Calculate z-test statistic: $$z=\frac{52,000-50,000}{5,000/\sqrt{30}}=\frac{2,000}{913.42}\approx 2.19$$

3. Critical value: For α=0.05 (one-tailed), $z_{0.05}=1.645$.

4. Decision:

   • Since $2.19>1.645$, reject $H_0$.
   • p-value: $P(Z>2.19)\approx 0.0143$ (from z-table).
   • Since $0.0143<0.05$, reject $H_0$.

Conclusion: There is statistically significant evidence that eSewa’s transaction volume increased after the update.

FIGURE 4: Z-Test for eSewa Transactions

figure:

graph LR
    A["Sample Mean (After): 52,000"] --> B["Population Mean (H₀): 50,000"]
    B --> C["Standard Error: 913.42"]
    C --> D["z = 2.19"]
    D --> E["Critical z: 1.645"]
    E -->|"z > critical"| F["Reject H₀"]

P-values and Decision Rules

• P-value: Probability of observing data as extreme as the sample, assuming $H_0$ is true.
  • If $p\le \alpha$, reject $H_0$.
  • If $p>\alpha$, fail to reject $H_0$.

Example Interpretation:

• For eSewa’s test, $p=0.0143<0.05$ → reject $H_0$.
• For Daraz’s CI example, since 48 hours is within the CI, we fail to reject the claim at 95% confidence.

FIGURE 5: P-value Decision Rule

figure:

graph LR
    A["Calculate p-value"] --> B["Compare to α"]
    B --> C["If p ≤ α"]
    C --> D["Reject H₀"]
    B --> E["If p > α"]
    E --> F["Fail to reject H₀"]

3. Type I and Type II Errors

Trade-off:

• Lowering $\alpha$ (e.g., 0.01) reduces Type I errors but increases Type II errors.
• Increasing sample size ($n$) reduces both errors.

FIGURE 6: Type I vs. Type II Errors

figure:

graph TD
    A["H₀ is True"] --> B["Correct Decision: Fail to Reject H₀"]
    A --> C["Type I Error: Reject H₀"]
    D["H₀ is False"] --> E["Type II Error: Fail to Reject H₀"]
    D --> F["Correct Decision: Reject H₀"]

Worked Example: NEPSE Stock Returns

NEPSE claims its stock returns are normally distributed with μ=12%. A sample of 40 stocks has a mean return of 10.5% and σ=4%. Test if the true mean is 12% at α=0.05.

1. State hypotheses:

   • $H_0$: $\mu =12\%$.
   • $H_a$: $\mu \ne 12\%$ (two-tailed).

2. Calculate t-test statistic (small sample, σ unknown): $$t=\frac{10.5-12}{4/\sqrt{40}}=\frac{-1.5}{0.632}\approx -2.37$$

3. Critical t-value (df=39, α=0.05/2): $t_{0.025}=\pm 2.023$.

4. Decision:

   • Since $-2.37<-2.023$, reject $H_0$.

Conclusion: The sample evidence suggests NEPSE’s stock returns are not normally distributed with μ=12% at 95% confidence.

FIGURE 7: T-test for NEPSE Returns

figure:

graph LR
    A["Sample Mean: 10.5%"] --> B["H₀ Mean: 12%"]
    B --> C["Standard Error: 0.632"]
    C --> D["t = -2.37"]
    D --> E["Critical t: ±2.023"]
    E -->|"t < -2.023"| F["Reject H₀"]

4. Chi-Square Tests: Categorical Data

Used to test:

• Goodness-of-fit: Does observed data fit an expected distribution?
• Independence: Are two categorical variables independent?

Example: Does Pathao’s surge pricing tier usage fit a uniform distribution?

FIGURE 8: Chi-Square Goodness-of-Fit

figure:

graph TD
    A["Observed Frequencies"] --> B["Expected Frequencies (Uniform)"]
    B --> C["Calculate χ² = Σ (O - E)²/E"]
    C --> D["Compare to Critical χ²"]
    D --> E["Decision: Fit or Not Fit"]

Worked Example: Daraz Order Categories

*Daraz’s orders are categorized as:

• Electronics (30%)
• Clothing (40%)
• Groceries (30%). A sample of 100 orders has: Electronics: 35, Clothing: 38, Groceries: 27. Test if the distribution fits Daraz’s claim at α=0.05.*

1. State hypotheses:

   • $H_0$: Observed distribution matches claimed (30%, 40%, 30%).
   • $H_a$: Distribution does not match.

2. Expected counts:

   • Electronics: $100\times 0.3=30$
   • Clothing: $100\times 0.4=40$
   • Groceries: $100\times 0.3=30$

3. Calculate χ²: $${\chi }^2=\frac{(35-30{)}^2}{30}+\frac{(38-40{)}^2}{40}+\frac{(27-30{)}^2}{30}=0.33+0.05+0.33=0.71$$

4. Critical χ² (df=2, α=0.05): $3.841$.

5. Decision:

   • Since $0.71<3.841$, fail to reject $H_0$.

Conclusion: The sample distribution does not significantly differ from Daraz’s claimed categories.

FIGURE 9: Chi-Square Calculation for Daraz

figure:

mermaid table

5. Real-World Applications

In the Real World

1. eSewa/Khalti: Transaction Fraud Detection

   • Idea: Hypothesis testing checks if fraud rates increase after new payment methods.
   • Example: eSewa tests if the mean fraud rate rises from 0.5% to 1% after adding UPI. A z-test compares the new sample mean to the old rate.

2. Ncell/NTC: Network Performance

   • Idea: Confidence intervals estimate call drop rates after network upgrades.
   • Example: Ncell claims its call drop rate is ≤2%. A 95% CI of [1.8%, 2.5%] includes 2%, so the claim is plausible.

3. Daraz: Inventory Management

   • Idea: Chi-square tests if product categories align with demand forecasts.
   • Example: Daraz checks if observed sales (Electronics: 35%, Clothing: 38%) match expected (30%, 40%). The χ²=0.71 suggests no significant deviation.

4. NEPSE: Stock Market Stability

   • Idea: t-tests verify if stock returns deviate from historical norms.
   • Example: If NEPSE’s sample mean return is 10.5% (vs. historical 12%), a t-test may reject the null hypothesis, signaling instability.

FIGURE 10: Real-World Examples Summary

figure:

mindmap
  root((Statistical Inference))
    eSewa: Hypothesis Testing for Fraud Rates
    Ncell: Confidence Intervals for Call Drop Rates
    Daraz: Chi-Square for Category Fit
    NEPSE: t-Tests for Stock Return Stability

Exam Tips

1. Master the Steps:

   • Always state $H_0$ and $H_a$ clearly.
   • For z/t-tests, show the formula and plug in numbers.
   • For chi-square, build the table with O, E, and $(O-E{)}^2/E$.

2. Interpret Results Correctly:

   • "Reject $H_0$" means evidence supports the alternative.
   • "Fail to reject" does not mean "prove $H_0$"; it means insufficient evidence.

3. Choose the Right Test:

   • Z-test: Large $n$, σ known.
   • t-test: Small $n$ or σ unknown.
   • Chi-square: Categorical data.

4. Show Work for Full Marks:

   • Write out calculations (e.g., standard error, test statistic).
   • Include p-values or critical values with decisions.

5. Real-World Linkage:

   • Connect tests to business scenarios (e.g., "How would a bank use a t-test for loan defaults?").
   • Use examples from the unit (e.g., Daraz, eSewa) to explain concepts.

6. Common Pitfalls:

   • Directionality: One-tailed vs. two-tailed tests.
   • Assumptions: Normality for t-tests, independence for chi-square.
   • Sample Size: Use z for $n>30$, t otherwise.

FIGURE 11: Exam Checklist

figure:

graph LR
    A["State H₀ and H₀"] --> B["Choose α"]
    B --> C["Calculate Test Statistic"]
    C --> D["Compare to Critical Value or p-value"]
    D --> E["Decision: Reject/Fail to Reject"]
    E --> F["Interpret in Context"]
    F --> G["Show All Work"]