IB Mathematics SL 4.9 The normal distribution -AI HL Paper 1- Exam Style Questions- New Syllabus
▶️ Answer/Explanation
The reaction time is normally distributed as
\(T\sim N(5,1^2)\).
(a)
We need to calculate
\(P(T<4)\).
Using the normal cumulative distribution function with mean \(5\) and standard deviation \(1\),
\(P(T<4)=0.158655\ldots\)
\(\therefore P(T<4)\approx 0.159\).
✅ Answer: \(0.159\)
(b)(i)
The slowest participants take the longest time to complete the test. Therefore, the required \(15\%\) is represented by the right-hand tail of the normal distribution.
Shade the region under the curve to the right of the cutoff value \(k\), where \(k>5\).
✅ Answer: Shade the rightmost \(15\%\) of the distribution.
(b)(ii)
Let \(k\) be the least completion time of a participant who will receive training.
Since the slowest \(15\%\) are selected,
\(P(T>k)=0.15\).
Equivalently,
\(P(T<k)=0.85\).
Using the inverse normal function with mean \(5\), standard deviation \(1\) and cumulative probability \(0.85\),
\(k=\operatorname{invNorm}(0.85,5,1)\)
\(k=6.03643\ldots\)
\(k\approx 6.04\text{ seconds}\).
This means that participants taking approximately \(6.04\) seconds or longer are among the slowest \(15\%\).
✅ Answer: \(6.04\) seconds