1. ## Help please midterm tomorrow!

A machine fills containers with a mean weight per container of 16.0 oz. If no more than 5% of the containers are to weigh less than 15.8 oz, what must the standard deviations of the weight equal?

Any help will be much appreciated!

2. What is the total distance between 15.8 and 16? On a normal curve, what is the score that corresponds to a chance of 0.05? That should help you get started, or jumpstart whatever work you have done so far. Have you tried anything - your other problem mentions "normal" curve, so I would imagine this is a "normally distributed" machine.

3. This is all the information he gave for the problem, he wasnt very clear :/ . Yes, I think this problem has a normal distribution.

4. Since the question asks about placing a bound on the variance ( $stdev^2$) given a mean and a probability, I immediately thought of CHEBYSHEV. (Note usually, Chebyshev is used via a given mean and variance, and one places bounds on the probability.)

While he provides bounds on the probabilities of random variables of any distribution given a mean and a variance, since we know the probability and the mean we can effectively place bounds on the variance.

So first of all here is the statement of his inequality:

$
If \; E[X] = \mu, \; Var(X) = \sigma^2, \;then\;for\;any\;a>0,$

$P({X\geq\mu+a})\leq\frac{\sigma^2}{\sigma^2+a^2}$
$P({X\geq\mu-a})\leq\frac{\sigma^2}{\sigma^2+a^2}$

So in our case, define the random variable $X$ to be difference between 16oz and the realized weight of the container. Clearly, given the wording of the problem, we may take $E[X]=0$
In our case we want
$
P({X>(0+(16-15.8)})<.05$

$Solving\;for\;\sigma\; \Rightarrow$

$(.05)=\frac{\sigma^2}{\sigma^2+.2^2}$

$\Rightarrow$

$\sigma^2=.010526\;or\;\sigma = .102598$

Note that if you use the Normal distribution as an approximation of the distribution of your random variable, you get $\sigma<.12159$

Chebyshev delivers the tighter bounds. Intuitively this makes sense, since it makes no assumption about the distribution of the random variable, other than its having finite moments.