Highschool (year 12) homework question (statistical analysis) I am getting stuck on part b of the following question
A company manufactures bricks with normally distributed weights. The mean weight is 0.96kg with standard deviation 0.045kg
a) Let Y be the mean of weight of 9 such bricks. Find the distribution of Y
b) What is the probability that a random sample of 9 bricks has mean weight more than 1kg? 
So my answer for a is
$Y \sim N(0.96, 0.045^{2})$
However, I don't know where to start for b). Could someone please help me with this? Thank you in advance 
 A: I will show you the general method for such a problem
and let you plug in the specific numbers for your particular problem.
Suppose bricks have weights distributed as
$\mathsf{Norm}(\mu, \sigma).$ [Notice that in this notation the second parameter is the standard deviation, not the variance.]
(a) The mean $Y = \bar X$ of $n = 9$ such bricks
is distributed as $\bar X \sim \mathsf{Norm}(\mu, \sigma/\sqrt{n}).$  [That is, $SD(\bar X) = \sigma/\sqrt{n}.]$
(b) You seek 
$$P(\bar X > a) = 
P\left(\frac{\bar X - \mu}{\sigma/\sqrt{n}} >
            \frac{a - \mu}{\sigma/\sqrt{n}}\right)
= P\left(Z > \frac{a - \mu}{\sigma/\sqrt{n}}\right),$$
where $Z$ has a standard normal distribution.
In the final expression, let $a = 1, \mu=0.96, \sigma = 0.045, n = 9.$ and use a printed table of the standard
normal cumulative distribution function to get your answer.
Using R, in which $\mathtt{pnorm}$ is a normal CDF with
specified parameters, one can get the numerical answer $P(\bar X > 1) = 0.00383$ as shown below. (Because of rounding, your result using printed normal tables will be
slightly less precise.)
n = 9; mu = 0.96; sg = 0.045
1 - pnorm(1, mu, sg/sqrt(n))  # last param is SD
[1] 0.003830381

In the following picture, the dotted density curve is for the
population of bricks, the solid curve is for the mean of $n = 1$ bricks, and the (small) area under the solid curve to the right of the vertical line represents the desired probability.

Notes: (1) Roughly speaking the PDF of the population is three times
as wide as the PDF for the PDF of the mean of nine. Thus
in order to include total probability 1 under the curve,
the PDF of the mean must be three times as tall.
(2) R is excellent statistical software, available free of charge for Windows, Mac and Unix computers from here. It is easy to learn to use--provided you use just what you need at each step.
