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
Best Answer
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.)
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.