Question

How to solve question 19 thanks

Answer 1

Let $X$ be the random variable for the weight of any given can, and let $\mu$ and $\sigma$ be the mean and standard deviation, respectively, for the distribution of $X$.

You have

$\begin{cases}\mathbb P(X377)=0.025\end{cases}$

Recall that for any normal distribution, approximately 99.7% of it lies within three standard deviations of the mean, i.e. $\mathbb P(\mu-3\sigma$. This means 0.3% must lie outside this range, $\mathbb P(X\mu+3\sigma)=0.003$. Because the distribution is symmetric, it follows that $\mathbb P(X$.

Also recall that for any normal distribution, about 95% of it falls within two standard deviations of the mean, so $\mathbb P(\mu-2\sigma$, which means 5% falls outside, and by symmetry, $\mathbb P(X>\mu+2\sigma)=0.025$.

Together this means

$\begin{cases}\mu-3\sigma=347\\\mu+2\sigma=377\end{cases}$

Solving for the mean and standard deviation gives $\mu=365$ and $\sigma=6$.