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    Use Both Normal Approximation and Poisson Approximation

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    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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    Solution Summary

    The solution gives detailed steps on finding the probability using both normal approximation and Poisson approximation.