Practice section 4 of 7

A manufacturer produces margarine packets with normally distributed mass, mean 502 g and standard deviation 0.8 g. Find the proportion containing less than 500 g.

The mean is increased to μg and the standard deviation is unchanged. Only 1 packet in 1000, on average, now contains less than 500 g. Find μ correct to 1 decimal place.

A grading machine rejects apples as too large if their mass exceeds 150 g, and too small if below 110 g. Over a long period 10% are too large and 15% too small. Apple mass in grams may be modelled by $N(\mu,\sigma^2)$. Find μ and σ.

$X\sim N(\mu,\sigma^2)$ and $P(X<85)=P(X>101)=0.0548$. Find μ and σ.

X is normally distributed with mean 10 and standard deviation 2.

Find $P(9<X<13)$ and $P(X<12)$.

Find the possible values of a if the probability that X lies between a and 12 is 0.15.

If three values are taken at random, find the probability that two are less than 12 and the other greater than 12.

X and Y are independent continuous normal random variables with means 10,12 and standard deviations 2,3 respectively. Find

(a) $P(Y<10)$,

(b) $P(2X>25)$,

(c) $P(Y<X)$,

(d) $P(4X+5Y>90)$,

(e) x such that $P(X_1+X_2>x)=0.25$, for independent observations $X_1,X_2$ of X.

Type A rod lengths are normally distributed with mean 5 cm and standard deviation 0.5 cm; type B lengths have mean 10 cm and standard deviation 1 cm. Find the probability that

(a) the total length of 3 type A rods and 2 type B rods lies between 33 and 36 cm,

(b) the total length of 6 type A rods exceeds three times the length of one randomly chosen type B rod.