HELPP Stats work

(a) State the conditions under which the normal distribution may be used as an appoximation to the binomial
distribution X ~ B(n, p).
(2)
(b) Write down the mean and variance of the normal approximation to X in terms of n and p.
(2)
A manufacturer claims that more than 55% of its batteries last for at least 15 hours of continuous use.
(c) Write down a reason why the manufacturer should not justify their claim by testing all the batteries they
produce.
(1)
To test the manufacturer’s claim, a random sample of 300 batteries were tested.
(d) State the hypotheses for a one-tailed test of the manufacturer’s claim.
(1)
(e) Given that 184 of the 300 batteries lasted for at least 15 hours of continuous use a normal approximation to
test, at the 5% level of significance, whether or not the manufacturer’s claim is justified.
(11 marks)



4
The amount of time in minutes a person has to wait on hold to a helpline is modelled by the Normal distribution X
with mean �, and standard deviation �
The probability a person spends less than 7 minutes waiting is 0.08076 and the probability a person spends less than 9
minutes waiting is 0.1587
(a) Showing your working clearly, find � and � to the nearest minute.

(b) A manager claims that the probability someone has to wait longer than 20 minutes is less than 5%. Using your
answer to part a, show that this claim is not correct.

(c) Assuming that the spread of the distribution remains the same as in part (a), calculate the largest mean waiting
time for which the probability someone has to wait longer than 20 minutes is less than 5%.
(8 marks)