Here are the times (in minutes) that each of these 15 coffee drinking students w

Question

Here are the times (in minutes) that each of these 15 coffee drinking students was able to stay awake:

30, 60, 45, 80, -5 (fell asleep before the lecture even started), 90, 100, 90, 115, 120, 80, 100, 80, 80, 100

After the lecture you decide that If you can stay awake for at least 90 of the 120 minutes the lecture lasts, you will get enough out of it to pass the course (this may be a bad criterion, but it’s the one you have decided to use).

OK, now what statistical analysis do you perform on this data? Let’s go with CONFIDENCE INTERVALS. Explain why this is a good method and then CALCULATE the 90% and 95% confidence intervals for the time these 15 coffee drinkers can stay awake on average.

Bottom line: Is coffee the key to success in a f2f Stat 200 class? (Would it be necessary in our on-line course?)

FINALLY, what are some of the flaws in this observational analysis? How might it be better designed?

Explanation / Answer

90% CONFIDENCE:

Note that              
              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.05          
X = sample mean =    77.66666667          
t(alpha/2) = critical t for the confidence interval =    1.761310136          
s = sample standard deviation =    33.26659987          
n = sample size =    15          
df = n - 1 =    14          
Thus,              
              
Lower bound =    62.53807088          
Upper bound =    92.79526245          
              
Thus, the confidence interval is              
              
(   62.53807088   ,   92.79526245   ) [ANSWER]

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95% CONFIDENCE:

Note that              
              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    77.66666667          
t(alpha/2) = critical t for the confidence interval =    2.144786688          
s = sample standard deviation =    33.26659987          
n = sample size =    15          
df = n - 1 =    14          
Thus,              
              
Lower bound =    59.24423771          
Upper bound =    96.08909562          
              
Thus, the confidence interval is              
              
(   59.24423771   ,   96.08909562   ) [ANSWER]

*********************

As coffee drinkers are not necessarily awake at least 90 minutes (part of these intervals are below 90 minutes), then coffee is not the key to f2f Stat 200 success.

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As we can see, the sample size is limited. Also, the participants might not be independent, as some of them might be close friends. Maybe we can try obtaining students from other classes as well.

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