Normal probabilities from context
Big idea: For a normal variable, probability is area under the curve. So P(X < a) means the area to the left of a.
Example: if test scores are normal, P(score < 70) means the proportion of students scoring below 70.
How to calculate common normal probabilities
Worked example
Let X~N(100,152). Find P(X<120).
Step by step
- Write formula first: z=(x-mean)/sd
- Use table/calculator to get P(Z<1.33)
- P(Z<1.33)≈0.908
Final answer
P(X<120)≈0.908.
| Target | Calculator setup |
|---|---|
| P(X<a) | normalcdf(-1E99,a,mean,sd) |
| P(X>a) | normalcdf(a,1E99,mean,sd) |
| P(a<X<b) | normalcdf(a,b,mean,sd) |
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Inverse normal: finding cut-off values
When inverse is needed: If the probability is given and x is unknown, use inverse normal.
Worked example
Heights are X~N(170, 62). Find the height exceeded by top 10%.
Step by step
- Top 10% means left area = 0.90
- Use invNorm(0.90,170,6)
- x≈177.7 cm
Final answer
Top 10% cut-off is about 177.7 cm.
Exam wording: "Exceeded by top 10%" means use cumulative 0.90, not 0.10.
Exam traps and past-paper habits
Common mistakes
- Using X directly without standardizing
- Confusing upper-tail with lower-tail
- Rounding too early
Safe method
- Standardize or use correct normalcdf inputs
- Sketch tail before calculating
- Round only at final line
Exam Tips:
- Write distribution first: X~N(mean,sd2).
- Mark the required tail with a quick sketch.
- State final probability to 3 s.f. unless told otherwise.