Constructing a confidence interval question

Question

I'm trying to solve this question but I'm having trouble figuring how to solve it or even start

I know that the final answer is (739.8028 ml, 746.1972 ml) but I'd really appreciate any help on how to get there? Thank you!

Answer 1

The calculation of the confidence interval of $ 95 \gamma $ percent is generally based on the pivotal quantity, which is a function (in this case) that depends on the random sample and the parameter of interest, that is $ \mu $ .

So the pivotal quantity for this case is $$ Q(x_{1},x_{2},...,x_{n},\mu)=\frac{\bar{x} -\mu} {\sigma/\sqrt n}\sim N(0,1).$$ Thus, Q is the pivotal quantity whose standard normal distribution does not depend on $\mu $, then there are $q_{1},q_{2} $ such that $P(q_{1}<Q <q_{2}) = \gamma $, for $ \gamma $ given in the interval $ (0,1). $

From the above, clearing the parameter of interest, that is $ \ mu $, we have that \ $ P (\bar {x} - \dfrac {\sigma} {\sqrt {n}} q_ {2} <\mu <\bar {x} + \dfrac {\sigma} {\sqrt {n}}q_ {2}) =\gamma $
For the symmetry of the distribution we can take $q_{1} = - q_{2} $ Since it is $ 95 $ percent, for the complement there is 0.05 percent that must be divided by 2, that is, 0.025, then looking in a standard normal table for 0.975 we have that $ q_ {2} = 1.96.$

$ \bar {x} = 743, \sigma = 6, n = 16.$ You plug in and find the 95 percent confidence interval.