Question

Approximately 85% of applicants get their G1 driver's license the first time they try the test. 80 applicants are scheduled to take the next test. The random variable X = The number of successful applicants during one session of administering the test.

a) Can this situation be approximated by a normal distribution? Explain. (2)

b) Using an approximation to the normal distribution, calculate the probability that exactly 65 applicants will pass the test (4)

c) Using the binomial distribution, calculate the probability that exactly 65 applicants will pass the test (2)

d) Using an approximation to the normal distribution, calculate the probability that more than 10 applicants will need to retake the test (4)

e) Can d) be answered using the binomial distribution? If so, which method (binomial distribution, approximation to the normal distribution) would you suggest someone use to find the answer? Explain. (2)

Worth 14 marks in total

Answer 1

a) Yes, the situation can be approximated by a normal distribution. For a normal distribution, we need a mean and variance.

For a binomial distribution, mean and variance are given by

  

b) Mean of the random variable X is given by

and

The probability that exactly 65 applicants will pass the test is given by

Using the continuity correction for discrete random variables, we have

Hence, the probability that exactly 65 applicants will pass the test = 0.080

c) Using the binomial distribution, the probability that exactly 65 applicants will pass the test

d) The probability that more than 10 applicants will need to retake the test

e) Yes, question d can be solved using the binomial distribution

This probability will need to be calculated by any binomial probability calculator.

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