Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Sections 7.1–7.2
Chapter Summary
The normal distribution — the bell curve — is the most important probability model in statistics. Heights, blood pressure, measurement errors, and countless other variables pile up near the middle and taper off symmetrically, and the normal curve describes that pattern with just two numbers: the mean μ (where the peak sits) and the standard deviation σ (how spread out it is). This chapter teaches you to find probabilities as areas under the curve and, in reverse, to find the value that cuts off a given area.
The key insight is that every normal distribution can be converted to the standard normal (mean 0, SD 1) using the z-score formula you already know from Chapter 3: z = (x − μ)/σ. Once you’re in z-scores, one table — or one StatCrunch calculator — answers every question. Going the other direction (finding x from a probability) is how real decisions get made: a hospital might need the cholesterol cutoff for the top 10% of patients, or a manufacturer the fill weight that only 1% of bottles fall below. Realistic example: if SAT-style scores are normal with μ = 500 and σ = 100, the score needed for the top 5% is found by getting z = 1.645 for area 0.95, then x = 500 + 1.645(100) ≈ 665. This “work backward” skill also sets up confidence intervals and hypothesis tests, where critical values like 1.96 come from exactly this process.
Study Guide
Key terms and definitions
- Normal distribution — symmetric, bell-shaped, defined by μ and σ; total area under the curve = 1.
- Standard normal distribution — the normal distribution with μ = 0 and σ = 1; values are z-scores.
- z-score conversion: z = (x − μ) ÷ σ; converts any normal value to the standard normal scale.
- Inverse normal problem — given an area (probability), find the value x (or z) that produces it.
- Notation: z_α is the z-score with area α to its right (e.g., z_0.05 = 1.645).
Essential formulas
- z-score: z = (x − μ) ÷ σ.
- Back-conversion: x = μ + zσ.
- Probability as area: P(a < X < b) = area under the normal curve between a and b.
- Common critical values: z_0.05 = 1.645, z_0.025 = 1.96, z_0.01 = 2.33, z_0.005 = 2.575.
Step-by-step procedures
- Find a probability P(a < X < b): (1) convert a and b to z-scores; (2) find the cumulative area to the left of each z (table or StatCrunch); (3) subtract: area(left of b) − area(left of a).
- Find x given a probability (“top 10%”): (1) translate the words into an area — “top 10%” means area 0.10 to the right, so cumulative area 0.90 to the left; (2) find the z with that cumulative area; (3) convert back: x = μ + zσ.
- Sketch first: always draw the bell curve, mark μ, shade the region. The sketch catches sign errors before they happen.
Common mistakes to avoid
- Forgetting to convert x to z before using the standard normal table — raw values and z-scores are not interchangeable.
- Mixing up “area to the left” and “area to the right.” Standard normal tables give area to the left; “top 10%” means area 0.10 on the right (cumulative 0.90).
- For “between” probabilities, computing only one area instead of subtracting two.
- Assuming data are normal without checking — the normal model only works if the data are actually bell-shaped (check a histogram first).
Self-check: I can…
- [ ] State the properties of a normal distribution and the standard normal.
- [ ] Convert between x-values and z-scores in both directions.
- [ ] Find probabilities for normal variables (less than, greater than, between).
- [ ] Find the value corresponding to a given percentile or “top/bottom %” cutoff.
- [ ] Explain what z_α notation means and recall the common critical values.
StatCrunch How-To
- Normal probabilities: Stat > Calculators > Normal — enter the mean and SD. Choose the direction:
<(less than),>(greater than), or Between. Enter the value(s) and click Compute; StatCrunch shades the area and gives the probability. - Inverse normal (find x from a probability): in the same Normal calculator, enter the probability in the probability box instead, select the direction, and click Compute — StatCrunch returns the x (or z) value.
- Standard normal: just set mean = 0 and SD = 1 in the Normal calculator.
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