Factory $A$ produces nuts whose mass may be assumed to be normally distributed with mean $\mu $

Question

Factory $A$ produces nuts whose mass may be assumed to be normally distributed with mean $\mu $ grams and standard deviation $\sigma $ grams. A random sample of $50$ nuts is taken. It is given that the probability that the mean mass is less than $247$ grams is $0.018079$, correct to $5$ significant figures. It is also given that the probability that the total mass exceeds $12600$ grams is $0.78397$, correct to $5$ significant figures. Find the values of $\mu $ and $\sigma $, giving your answers to the nearest grams. [5] (For this question, you should state clearly the values of the parameters of any normal distribution you use.) Factory $B$ produces bolts and nuts. The masses, in grams, of bolts and nuts produced are modelled as having independent normal distributions with means and standard deviations as shown in the table: Mean Mass (in grams) Standard Deviation (in grams) Bolts $745$ $7.3$ Nuts $250$ $5$ Find the probability that the mass of a randomly chosen bolt differs from $3$ times the mass of a randomly chosen nut by at least $40$ grams. [4] This factory introduces a new process which is able to reduce the mass of each nut by $10\%$. Find the probability that the total mass, after the introduction of this process, of $10$ randomly chosen nuts is less than $2.24$ kg. [3]

Accepted Answer

(i) $\mu=255\text{ g},\ \sigma=27\text{ g}$ (ii)(a) $0.0214\text{ (3 s.f.)}$ (ii)(b) $0.241\text{ (3 s.f.)}$

	Mean Mass (in grams)	Standard Deviation (in grams)
Bolts	\(745\)	\(7.3\)
Nuts	\(250\)	\(5\)

2017 MJC P2 Q10

2017 MJC P2 Q10

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