| Exam Board | OCR MEI |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2009 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two or more different variables |
| Difficulty | Standard +0.8 This question requires understanding of linear combinations of normal variables across multiple parts with increasing complexity. Parts (i)-(ii) are routine standardization, but part (iii) requires combining 7 shelves and 6 gaps (13 variables total) and then applying binomial probability, while part (iv) involves finding the distribution of the difference between two unit heights. The multi-step nature, need to track variance calculations carefully, and the final inverse normal calculation make this moderately challenging but still within standard S3 scope. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04b Linear combinations: of normal distributions |
1 Andy, a carpenter, constructs wooden shelf units for storing CDs. The wood used for the shelves has a thickness which is Normally distributed with mean 14 mm and standard deviation 0.55 mm . Andy works to a design which allows a gap of 145 mm between the shelves, but past experience has shown that the gap is Normally distributed with mean 144 mm and standard deviation 0.9 mm . Dimensions of shelves and gaps are assumed to be independent of each other.\\
(i) Find the probability that a randomly chosen gap is less than 145 mm .\\
(ii) Find the probability that the combined height of a gap and a shelf is more than 160 mm .
A complete unit has 7 shelves and 6 gaps.\\
(iii) Find the probability that the overall height of a unit lies between 960 mm and 965 mm . Hence find the probability that at least 3 out of 4 randomly chosen units are between 960 mm and 965 mm high.\\
(iv) I buy two randomly chosen CD units made by Andy. The probability that the difference in their heights is less than $h \mathrm {~mm}$ is 0.95 . Find $h$.
\hfill \mbox{\textit{OCR MEI S3 2009 Q1 [18]}}