Standard +0.3 This is a standard S3 question on linear combinations of normal random variables. Students must form 3X and 2Y, find their distributions using standard results (mean and variance of sums), then compute P(3X < 2Y) by considering the distribution of 2Y - 3X. The method is routine and well-practiced, requiring only careful arithmetic and normal table lookup, making it slightly easier than average.
5 The duration, \(X\) minutes, of a timetabled 1-hour lesson may be assumed to be normally distributed with mean 54 and standard deviation 2.
The duration, \(Y\) minutes, of a timetabled \(1 \frac { 1 } { 2 }\)-hour lesson may be assumed to be normally distributed with mean 83 and standard deviation 3.
Assuming the durations of lessons to be independent, determine the probability that the total duration of a random sample of three 1 -hour lessons is less than the total duration of a random sample of two \(1 \frac { 1 } { 2 }\)-hour lessons.
(7 marks)
5 The duration, $X$ minutes, of a timetabled 1-hour lesson may be assumed to be normally distributed with mean 54 and standard deviation 2.
The duration, $Y$ minutes, of a timetabled $1 \frac { 1 } { 2 }$-hour lesson may be assumed to be normally distributed with mean 83 and standard deviation 3.
Assuming the durations of lessons to be independent, determine the probability that the total duration of a random sample of three 1 -hour lessons is less than the total duration of a random sample of two $1 \frac { 1 } { 2 }$-hour lessons.\\
(7 marks)
\hfill \mbox{\textit{AQA S3 2007 Q5 [7]}}