An energy drink company routinely measures the caffeine cont

An energy drink company routinely measures the caffeine content of their drinks. Suppose that the complay selects 100 samples of the drink every hour from it\'s production line and determines the caffeine content. From previous analyses, the company is confident that the caffeine content has a normal distribution with a = 6.8mg. During a one hour period, the 100 samples yielded a mean caffeine content of x = 130mg Construct a 95% confidence interval for the mean caffeine content mu of the energy drink produced during the hour in which the samples were taken. What would happen to the width of the of the confidence interval if our confidence level changed for this interval went from 95% to 99%? Explain why this makes sense. What would happen to the width of the confidence interval if we were to double sample size from 100 to 200? Explain why this makes sense (No need to construct a new confidence interval). Suppose the company repeats this process every hour of every day for one week, thus constructing 40 separate confidence intervals for the estimated mean caffeine content of their drinks. Of these intervals, how many would you expect to fail to contain the value of mu thus providing an inaccurate estimation of the mean caffeine content?

Solution

a)

Note that              
              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    130          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    6.8          
n = sample size =    100          
              
Thus,              
              
Lower bound =    128.6672245          
Upper bound =    131.3327755          
              
Thus, the confidence interval is              
              
(   128.6672245   ,   131.3327755   ) [ANSWER]

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b)

It will be WIDER if we make it 99%. This makes sense because if we want to be \"more confident\" that we have the true mean, we must expand our interval.

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c)

If the new sample size is 200, th new interval will be NARROWER. This makes sense because the sample size increased, hence we know more about the population. Thus, for a narrower interval, we are as confident as before.

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d)

We are 95% confident that we have the true mean, so we are 5% \"unconfident\" that we don\'t have it.

Thus, 5% of 40 = 2. So we expect about 2 intervals to fail to accurately estimate u. [ANSWER, 2]