Mean/SD and bell-shaped symmetry; Emergence as sums of independent random contributions; 68:95:99.7 rule; Non-standard probabilities using technology; Inverse normal/quantiles and contextual predictions
Example 1
X is normal with mean 50 and standard deviation 3. Find P(X<53). Round to 4 decimal places.
Show worked answer
Answer: 0.8413
- Standardise z=σx−μ=1, then evaluate Φ(1).
- Rounded result: 0.8413.
Example 2
Z is standard normal. Find P(−1<Z<1). Round to 4 decimal places.
Show worked answer
Answer: 0.6827
- Subtract the lower cumulative probability from the upper.
- Rounded result: 0.6827.
Example 3
X is normal with mean 40, standard deviation 5. Find its 90th percentile. Round to 2 decimal places.
Show worked answer
Answer: 46.41
- Find the standard-normal 0.90 quantile and transform x=μ+σz.
- Rounded result: 46.41.