Standard scores, symmetry and normal probabilities.
Example 1
A normal distribution has mean 60 and standard deviation 5. Find the z-score for x=85.
Show worked answer
Answer: 5
- z=σx−μ.
- z=585−60=5.
Example 2
X is normally distributed with mean 40 and standard deviation 3. Find P(X>40).
Show worked answer
Answer: 0.5
- A normal distribution is symmetric about its mean.
- Half the area lies above the mean: probability 0.5.
Example 3
X is normal with mean 80 and standard deviation 6. Find P(X<92). Give your answer to 4 decimal places.
Show worked answer
Answer: 0.9772
- Standardise: z=692−80=2.
- Use the normal cumulative distribution at 2.
- Rounded answer: 0.9772.