Normal Distribution Probability Calculation
Use the standard normal distribution to find probabilities related to a normally distributed variable, including converting to z-scores and using the z-table.
Problem Scenario
The heights of adult women in a certain country are normally distributed with a mean of 164 cm and a standard deviation of 7 cm. Find: (a) the probability that a randomly selected woman is taller than 175 cm, and (b) the probability that a woman's height is between 155 cm and 172 cm.
Given Data
Requirements
- Convert raw scores to z-scores using the z-score formula
- Use the standard normal distribution to find the requested probabilities
Solution
Step 1:
Part (a): Convert 175 cm to a z-score. z = (x - mu) / sigma = (175 - 164) / 7 = 11 / 7 = 1.571. Round to z = 1.57.
Step 2:
Find P(X > 175) = P(Z > 1.57) = 1 - P(Z < 1.57). From the z-table, P(Z < 1.57) = 0.9418. Therefore P(X > 175) = 1 - 0.9418 = 0.0582.
Step 3:
Part (b): Convert both values to z-scores. For 155 cm: z_1 = (155 - 164) / 7 = -9 / 7 = -1.286, round to z = -1.29. For 172 cm: z_2 = (172 - 164) / 7 = 8 / 7 = 1.143, round to z = 1.14.
Step 4:
Find P(155 < X < 172) = P(-1.29 < Z < 1.14) = P(Z < 1.14) - P(Z < -1.29). From the z-table: P(Z < 1.14) = 0.8729 and P(Z < -1.29) = 0.0985. Therefore P(155 < X < 172) = 0.8729 - 0.0985 = 0.7744.
Step 5:
Interpretation: (a) About 5.82% of women are taller than 175 cm. (b) About 77.44% of women have heights between 155 cm and 172 cm.
Final Answer
(a) P(X > 175) = 0.0582 or about 5.82%. (b) P(155 < X < 172) = 0.7744 or about 77.44%. Roughly 1 in 17 women are taller than 175 cm, and over three-quarters of women have heights between 155 cm and 172 cm.
Key Takeaways
- ✓The z-score tells you how many standard deviations a value is from the mean. Positive z-scores are above the mean; negative z-scores are below.
- ✓For "between" probabilities, subtract the smaller cumulative probability from the larger one: P(a < X < b) = P(Z < z_b) - P(Z < z_a).
- ✓The empirical rule (68-95-99.7) provides quick approximations: about 68% of data fall within 1 SD, 95% within 2 SDs, and 99.7% within 3 SDs of the mean.
Common Errors to Avoid
- ✗Forgetting to subtract from 1 when finding upper-tail probabilities. P(X > value) = 1 - P(X < value), because z-tables typically give left-tail (cumulative) probabilities.
- ✗Using the wrong sign for z-scores. Values below the mean produce negative z-scores; values above the mean produce positive z-scores. Double-check the subtraction.
- ✗Confusing P(Z < z) with P(Z > z). Always note whether the problem asks for "less than," "greater than," or "between" and use the correct tail(s).
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Common questions about this problem type
Round to the nearest value in your z-table (typically two decimal places). For more precision, use linear interpolation between adjacent table values. Alternatively, use a calculator or software that computes exact normal probabilities.
The normal distribution is appropriate when the data are approximately bell-shaped and symmetric. For clearly skewed, discrete, or bounded data, other distributions (e.g., binomial, Poisson, exponential) may be more appropriate. You can check normality with a histogram, Q-Q plot, or a formal test like Shapiro-Wilk.