Standardise, find probabilities and reverse normal calculations.
Example 1
X is normal with mean 30 and standard deviation 5. Find P(X>39). Give your answer to 4 decimal places.
Show worked answer
Answer: 0.0359
- z=539−30=1.8.
- The right-tail probability is 1−Φ(1.8).
- Rounded answer: 0.0359.
Example 2
X is normal with mean 20 and standard deviation 6. Find k such that P(X≤k)=0.95. Give your answer to 2 decimal places.
Show worked answer
Answer: 29.87
- Use inverse normal to find z with Φ(z)=0.95.
- Transform back: k=20+6z.
- Rounded answer: 29.87.
Example 3
X is normal with mean 30 and standard deviation 5. Find P(20<X<34.5). Give your answer to 4 decimal places.
Show worked answer
Answer: 0.7932
- The lower and upper z-scores are−2 and 0.9.
- Subtract cumulative probabilities: Φ(0.9)−Φ(−2).
- Rounded answer: 0.7932.