## Wine tasting from Prob and Stat for Engineering and the Sciences by Devore

OK, I am completely baffled at how to come up with a standard deviation on the following problem. This is my pathetic attempt to solve. Please help me figure out how to come up with the standard deviation which is the key to solving this. Problem 8.52 Oh, is the estimation of the mean correct or have I totally botched that too.

The article reported on an experiment to investigate whether wine tasters could distinguish between more expensive reserve wines and their regular counterparts. Wine was presented to tasters in four containers labeled A, B, C, and D, with two of these containing the reserve wine and the other two the regular wine. Each taster randomly selected three of the containers, tasted the selected wines, and indicated which of the three he/she believed was different from the other two. Of the n=855 tasting trials, 346 resulted in correct distinctions. Does this provide compelling evidence for concluding that tasters of this type have some ability to distinguish between the reserve and regular wines? State and test the relevant hypotheses using the P-value approach. Are you particularly impressed with the ability of tasters to distinguish between the two types of wine?

Since the taster has a 1/3 chance of randomly selecting the wine that is different from the other two, and you either get it right or wrong, we have a binomial distribution with p=.333 and q=.667. Out of 855 tasting trials, by random chance the mean should be 855/3=285. The null hypothesis should then be $H_{0}:\mu=285$ with $H_{a}:\mu>285$.

The closest I can come to a standard deviation is 346-285=61 and $61^2$ is 3721 divided by 854 gives 4.36.

Now take $\frac{346-285}{\frac{3721}{\sqrt{855}}}$=.479=z $\Phi(.479)$=.6844 P-value=2(1-.6844)=.6312.