Confidence Intervals and Hypothesis Testing

Confidence Intervals and Hypothesis Testing

Why These Tools Matter

Data analysts are constantly asked to draw conclusions from samples rather than entire populations. Two foundational statistical tools make that possible: confidence intervals and hypothesis tests. Both appear in the CompTIA Data+, Google Data Analytics, and IBM Data Science Professional credential exams, and both are skills you will use on the job every week.

---

Confidence Intervals

A confidence interval (CI) gives a range of plausible values for a population parameter (like a mean or proportion) based on sample data. It has three parts:

1. Point estimate — the sample statistic (sample mean x̄ or sample proportion p̂).

2. Margin of error (ME) — how far the interval extends on each side.

3. Confidence level — how often intervals built this way would capture the true parameter if sampling were repeated many times (common choices: 90%, 95%, 99%).

The general formula is:

```

CI = point estimate ± (critical value × standard error)

```

Standard error for a mean: SE = s / √n (use population σ if known; otherwise use sample s).

Standard error for a proportion: SE = √(p̂(1 − p̂) / n).

Critical values (z\*) for common confidence levels:

• 90% → z\* = 1.645
• 95% → z\* = 1.960
• 99% → z\* = 2.576

When the population standard deviation is unknown and the sample is small (commonly n < 30), use a t-distribution critical value instead, with degrees of freedom = n − 1.

Interpreting a CI correctly: A 95% CI does not mean there is a 95% probability the true mean is in this interval. It means that the method produces intervals that capture the true parameter in 95% of repeated samples. State it as: "We are 95% confident that the true population mean falls between [lower] and [upper]."

---

Hypothesis Testing — The Framework

Hypothesis testing is a formal decision procedure. You start with a default assumption and ask whether the data give enough evidence to reject it.

Step 1 — State the hypotheses.

• Null hypothesis (H₀): The "no effect" or "status quo" statement. Always contains an equality (=, ≤, or ≥).
• Alternative hypothesis (Hₐ): What you are trying to support. Can be two-tailed (≠), left-tailed (<), or right-tailed (>).

Step 2 — Choose significance level α. The most common choice is α = 0.05.

Step 3 — Select the test statistic.

• Use a z-test when σ (population SD) is known, or when n ≥ 30 (by the Central Limit Theorem).
• Use a t-test when σ is unknown and n < 30.

Step 4 — Compute the test statistic.

• z = (x̄ − μ₀) / (σ / √n)
• t = (x̄ − μ₀) / (s / √n)

Step 5 — Find the p-value and compare to α.

• If p-value ≤ α: reject H₀. The result is statistically significant.
• If p-value > α: fail to reject H₀. Insufficient evidence.

Step 6 — State a plain-language conclusion tied to the original question.

In Python, `scipy.stats` handles this cleanly:

```python

from scipy import stats

t_stat, p_value = stats.ttest_1samp(data_column, popmean=50000)

```

---

Common p-Value Misconceptions

The p-value is the probability of obtaining a test statistic at least as extreme as the observed one, assuming H₀ is true. It is NOT:

• The probability that H₀ is true.
• The probability that the result occurred by chance.
• A measure of practical importance or effect size.

A statistically significant result (small p) can be practically meaningless, especially with very large samples. Always pair statistical significance with effect size or business context.

---

Data Safety and Professional Practice

City employee salary data and similar government datasets are public records, but always verify the data use policy of any portal before downloading. When working with data that could be linked to individuals:

• Do not share raw data files outside your authorized environment.
• Follow your organization's data governance policy.
• In Python, avoid printing individual rows in shared notebooks; summarize instead.

While this is not a physical-hazard lab, data integrity is your equivalent of worksite safety: decisions made on incorrectly processed data can harm real people through misallocated budgets, discriminatory patterns, or flawed policy.

---

Putting It Together

For today's lab you will load a city employee salary CSV, state a claim about the mean annual salary, build a confidence interval, and run a one-sample t-test in Python. Each step maps directly to the six-step framework above. Keep your rubric (12 points, 75% to pass) in front of you: hypotheses, test selection, computation, and plain-language conclusion each carry points.