# Use Both Normal Approximation and Poisson Approximation

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.

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