a. Expand the expression (a + b) as a sum of terms in a and b.
b. Steel bolts are produced to a specified length. There is a probability of 5% that a bolt will fall outside of
this range. A sample of 7 bolts is extracted and measured. Assuming the bolt lengths are normally
distributed, use your answer to part (a) to calculate the probability that within the sample:
i. none are faulty
ii. exactly one is faulty
iii. one or less are faulty.
c. Based on the same sample as part (b):
i. Show that it is appropriate to assume a Poisson distribution to analyse this sample.
ii. Recalculate your answers to part (b) (i), (ii) and (iii) assuming a Poisson distribution.
iii. Comment on your results from this calculation comparing it with your answers to part (b).
d. Based on your answers to parts (b) and (c) above:
i. Comment on how effective you think sampling 7 bolts would be in terms of checking that 90% are
within the chosen range for a given batch taken from a production line.
ii. In the light of your answer to (i), state how you would improve the effectiveness of the sampling to
ensure greater confidence in the production process.
The solution gives detailed steps on finding the probability using both normal approximation and Poisson approximation.