Fizzy Waters promotes their alkaline water product for everyone on the basis that alkaline water is good for health as it neutralizes acids produced in the body. They boast having a mean alkalinity level of 50 mg/liter. Alkaline water has a higher pH level than regular drinking water and Fizzy Waters claims that its higher Hydrogen content provides better hydration than regular water. To test their claim, you contact Fizzy Waters and they allow you to collect samples from their manufacturing plant to test for yourself. You collect 100 random samples of their alkaline water and find that the mean and standard deviation are y = 32.2mg/liter and 14.4mg/liter. With 99% confidence, is there enough evidence to support their claim that the population mean exceeds 50 mg/liter?

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.