In this question, you should state clearly all the distributions that you use, together with the values of the appropriate parameters.

A company sells hand sanitiser in bottles of two sizes – small and large. The amounts, in (ml), of hand sanitiser in the small and large bottles, are modelled as having independent normal distributions with means and standard deviations as shown in the table.

(i) Find the probability that the amount of hand sanitiser in a randomly chosen small bottle is less than (100 ml).

[1]

(ii) During a quality control check on a batch of small bottles of hand sanitiser, 100 small bottles are randomly chosen to be inspected by an officer one at a time. Once he finds five bottles, each with amount of hand sanitiser less than (100 ml), that batch will be rejected. Find the probability that he had to check through all 100 bottles to reject that batch.

[2]

(iii) Given that the amount of hand sanitiser in (85 %) of the large bottles lie within 9 (ml) of the mean, find (sigma).

[3]

(iv) Given instead that (sigma=6), find the probability that the amount of hand sanitiser in a randomly chosen large bottle is less than five times the amount of hand sanitiser in a randomly chosen small bottle.

[3]

(i) $\mathrm{P}(X<100)=0.054799=0.0548$

(ii) $0.00880$ (3 s.f.)

(iii) $\sigma=6.25(3$ s.f. $)$