The idea, step by step
Turn the wording into an area
Sketch the situation or mark the requested interval before using technology. For P(X<a), calculate a lower-tail area; for P(X>a), use one minus the lower-tail area. For an interval, subtract two cumulative areas. Standardise with the model’s standard deviation, not its variance.
Give probabilities between zero and one, and distinguish a probability from an expected count. If n independent observations follow the same model, the expected count in an interval is n times its probability.
Inverse normal means work from area to value
For the pth lower-tail proportion, first find z=Φ⁻¹(p), then x=μ+zσ. “Top 10%” corresponds to lower-tail proportion 0.90, not 0.10. Check that an upper percentile lies above the mean.
Keep guard digits in z and round the final cutoff to sensible units. In a continuous model, including or excluding an endpoint does not change the probability. State any normal-model assumptions in the final conclusion.
A common trap
Use inverseNormal(0.10) for the top 10% cutoff.
Instead: The top 10% starts at the 90th percentile, so its lower-tail area is 0.90.
See the thinking
Between two masses
Mass X has mean 500 g and SD 4 g. Find P(496<X<506).
- The standardised limits are -1 and 1.5.
- Compute Φ(1.5)-Φ(-1)=0.7745.
- This is about 77.45% of packets under the model.
An upper cutoff
Scores are normal with mean 70 and SD 8. Find the cutoff for the highest 10%.
- Use lower-tail area 0.90.
- z=Φ⁻¹(0.90)≈1.2816.
- x=70+8z≈80.25.
Check your understanding
For X normal with mean 20 and SD 2, find the median.
Show answer
20.
The median splits the symmetric normal area in half and equals the mean.
Your turn
6 questions · 18 marks · Work at your own pace.
Read the questions online
- For Z standard normal, calculate P(Z<1.2).3 marks
- For Z standard normal, calculate P(Z>1.2).3 marks
- For X with mean 50 and SD 6, find P(44<X<62).3 marks
- For X with mean 100 and SD 12, find the 25th percentile.3 marks
- A normal quality model gives P(pass)=0.94. Out of 250 items, find the expected number passing.3 marks
- Find the upper 5% cutoff for a normal variable with mean 40 and SD 3.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- For Z standard normal, calculate P(Z<1.2).3 marks
Use the standard normal lower-tail function at 1.2.
0.8849. - For Z standard normal, calculate P(Z>1.2).3 marks
Use 1-Φ(1.2).
0.1151. - For X with mean 50 and SD 6, find P(44<X<62).3 marks
The z-limits are -1 and 2.
Subtract Φ(-1) from Φ(2).
0.8186. - For X with mean 100 and SD 12, find the 25th percentile.3 marks
Use z=Φ⁻¹(0.25).
Calculate 100+12z.
91.91. - A normal quality model gives P(pass)=0.94. Out of 250 items, find the expected number passing.3 marks
Expected count=250×0.94.
This is an expectation, not a guaranteed realised count.
235. - Find the upper 5% cutoff for a normal variable with mean 40 and SD 3.3 marks
Top 5% means lower-tail area 0.95.
Calculate 40+3Φ⁻¹(0.95).
44.93.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics: statistical models. Original lessons and assessment practice; not an official SACE assessment.
Official curriculum source ↗