Practice section 5 of 5

(i) Factory A’s nut mass is normal with mean $\mu$ grams and standard deviation $\sigma$ grams. For a random sample of 50, $P(\bar X<247)=0.018079$ and $P(\sum X_j>12600)=0.78397$, each correct to 5 s.f. Find $\mu,\sigma$ to the nearest gram. [5]

(ii) State clearly the parameters of every normal distribution used. Factory B’s independent mass models are:

Product	Mean (g)	Standard deviation (g)
Bolt	745	7.3
Nut	250	5

(a) Find the probability that a randomly chosen bolt’s mass differs from three times one randomly chosen nut’s mass by at least 40 g. [4]

(b) A new process reduces each nut’s mass by 10%. Find the probability that the total mass of 10 randomly chosen new nuts is below 2.24 kg. [3]

A bag has three identically shaped balls numbered 1,2,3. Draw one at a time at random with replacement. Let $X$ be the number of draws until any ball is drawn a second time; matching draws need not be consecutive.

(i) Show $P(X=4)=2/9$ and find the distribution of $X$. [3]

(ii) Show $E(X)=26/9$ and find exact $\operatorname{Var}(X)$. [3]

(iii) Using a suitable approximation, find the probability that the mean of 44 independent observations exceeds 3. [3]

In the problem’s model, birth weights for babies of non-smoking mothers at a hospital are normal with mean 3.05 kg and variance $\sigma^2$. The hospital’s stated classifications are:

Birth weight	Classification in the problem
Below 1.5 kg	Very low
1.5 kg to 2.5 kg	Low
2.5 kg to 4.0 kg	Normal
Above 4.0 kg	High

(i) A sample suggests 20.2% are in the low category. Assuming this holds for the population, find two possible $\sigma$ values to 2 dp and explain why one should be rejected. [4]

Babies of smoking mothers have mean 2.80 kg. For the remaining parts, explicitly assume both populations are normal with standard deviation 0.86 kg.

(ii) Three smoking mothers are chosen at random. Find the probability all their babies are outside the normal-weight category, stating a required assumption. [3]

(iii) Find the probability that the average weight of two babies of non-smoking mothers differs from twice the weight of one baby of a smoking mother by less than 2 kg. [3]

Babies outside the normal category remain in hospital for observation until stable. Depending on treatment and length of stay, hospitalisation cost per baby has mean $2800 and standard deviation $ 500.

(iv) For 50 such babies, find the probability that average cost exceeds $3000. [2]