Practice section 5 of 7

Pineapple masses in a large warehouse are normally distributed with mean 1.52 kg and standard deviation 0.070 kg. Six randomly chosen pineapples are packed in a box of mass 2.15 kg. Find the probability that the filled box has mass below 11.2 kg.

For 3 randomly chosen filled boxes, find the probability that at least 1 has mass below 11.2 kg.

The contents of a packet of cereal A have a normal mass distribution with mean 625 g and standard deviation 15 g. Packaging has an independent normal distribution with mean 25 g and standard deviation 3 g. Give answers to 3 significant figures.

(a) Find the probability that one packet of A has total mass exceeding 630 g.

(b) Find the probability that the contents of 4 random packets of A exceed 2450 g.

Cereal B contents have normal mass with mean 465 g and standard deviation 10 g. Find the probability that the contents of 4 random packets of B outweigh the contents of 3 random packets of A.

Chicken and turkey masses in kg have independent normal distributions.

Bird	Mean mass	Standard deviation
Chicken	2.2	0.5
Turkey	10.5	2.1

Chickens cost 3 dollars per kg and turkeys 5 dollars per kg.

(i) Find the probability that a random chicken costs more than 7 dollars.

(ii) Find the probability that both a random chicken costs more than 7 dollars and a random turkey more than 55 dollars.

(iii) Find the probability that their total price exceeds 62 dollars.

(iv) Explain why(iii) is greater than (ii).

Independent X,Y have distributions $N(2\mu,\sigma^2)$ and $N(3\mu,3\sigma^2)$ respectively, with σ>0. Let $Z\sim N(0,1)$.

(i) Given $P(X<2\mu+2\sigma^2)=k$, show that $P(Z<2\sigma)=k$.

(ii) Write down the mean and variance of Y−X. Hence find $P(|Y-X-\mu|>4\sigma^2)$ in terms of k.

$X\sim N(3,4)$, with mean 3 and variance 4. S sums 100 independent observations of X; T sums a further 300 independent observations. Give numerical answers to 3 decimal places.

(i) Find $P(S>310)$.

(ii) Find $P(3S>50+T)$.

(iii) N sums n independent observations of X. State the approximate value of $P(N>3.5n)$ as n becomes very large, with justification.

The times taken by two runners A and B to run 400 metre races are independent and normally distributed with means 45.0 seconds and 45.2 seconds and standard deviation 0.5 seconds and 0.8 seconds respectively. The two runners are to compete in a 400 metre race for which there is a track record of 44.5 seconds.

(i) Calculate the probability of runner A breaking the track record. Hence, show that the probability of runner B breaking the record is greater than that of runner A. [3]

(ii) Find the probability of runner A beating runner B. [3]

(iii) Find the probability that the total sum of four randomly chosen times of A is more than 4 times a randomly chosen time of B. [5]