Discussion questions · Section 1 of 3

Packages from a packing machine have a mass which is normally distributed with mean 200 g, and standard deviation 2 g. Find the probability that a package from the machine weighs

(i) between 198.5 and 199.5 g,

(ii) less than 197 g,

(iii) more than 200.5 g.

An examination is marked out of 100 and taken by a large number of candidates. The mean mark is 72.1 and standard deviation 15.2. Give a reason why a normal distribution with this mean and standard deviation would not give a good approximation to the marks.

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.