It is 1581 Anno Domini. At an Under graduate School, besides Assistant Academic

Question

It is 1581 Anno Domini. At an Under graduate School, besides Assistant Academic Director of Mathematics, I am also the school appointed CPA, Coffee Pot Attendant. It is very important office sponsored by the Holy See I have taken this job very seriously because believe that I am the key to increased productivity at the Undergraduate School. Why, by mid-morning, many of my colleagues act as if they were drained of their energy. It is imperative that I restore productivity via a secret naturally-occurring molecule, caffeine........ In order to see if my secret molecule works, a random sample of ten colleagues who had coffee before the Dean's meeting was selected. I have observed the time, in hours, for those 10 colleagues to stay awake at the extremely long-winded Dean's meeting as soon as it started. One fell asleep even before the meeting started. Now, I have to complete a report to the Provost's Office on the effectiveness of a secret molecule so that the undergraduate school can file for a patent at the United Provinces Patent and Trademark Office as soon as possible. The following information is needed. a) What is a 95% confidence interval for the time my colleagues can stay awake on average for all of my colleagues? b) Was my secret molecule effective in increasing their attention span and their staying awake? And, please explain.....

Explanation / Answer

a. The sample of ten employees is small, n<30 and random. The population standard deviation is unknown. Assume the time (in hours) to stay awake is normally distributed. Use, Student's t model with n-1 degrees of freedom to compute the confidence interval for 1-population mean.

The 95% c.i for the time the collegues can stay awake is as follows:

xbar+-t(s/sqrt n), where, xbar is sample mean, t is t critical at alpha/2, and n-1 degrees of freedom, s is sample standard deviation, and n is sample size. The computational formula for xbar: sigma x/n, and s=sqrt[1/n-1 sigma (x-xmu)^2].

=2.330+-2.262(2.002/sqrt 10)

=(0.898, 3.762) [ans]

b. The confidence interval does not ontain 0, therefore, the result is significant at alpha=0.05. One can conclude that the secret molecule is effective in increasing the attention span and their staying awake.

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