A company that manufacturers coffee for use in commercial machines monitors the
ID: 3047751 • Letter: A
Question
A company that manufacturers coffee for use in commercial machines monitors the caffeine content in its coffee. The company select 25 samples of coffee every hour from its production line and determines the caffeine content. From historical data, the caffeine content in milligrams is known to have a normal distribution. During a 1-hour time period, the 25 samples yielded an average caffeine content of 110 mg and standard deviation 7.1 mg. Use the sample information to calculate 90% and 96% confidence intervals for the mean caffeine content of the coffee produced during the hour in which the 25 sample were selected.
The 90% CI:
The lower limit is: The upper limit is:
he 96% CI:
The lower limit is: The upper limit is:
In the next hour, the company wishes to construct a 95% confidence interval of the average caffeine with margin of error m = 2.5 mg. How many samples of coffee should the company select? Remember to round your answer up to the next whole number.
In the next hour, the company wishes to construct a 99% confidence interval of the average caffeine with margin of error m = 3 mg. How many samples of coffee should the company select? Remember to round your answer up to the next whole number.
Explanation / Answer
(a)
n = 25
x-bar = 110
s = 7.1
% = 90
Standard Error, SE = s/n = 7.1/25 = 1.42
Degrees of freedom = n - 1 = 25 -1 = 24
t- score = 1.710882067
Width of the confidence interval = t * SE = 1.71088206673347 * 1.42 = 2.429452535
Lower Limit of the confidence interval = x-bar - width = 110 - 2.42945253476153 = 107.5705475
Upper Limit of the confidence interval = x-bar + width = 110 + 2.42945253476153 = 112.4294525
The 90% confidence interval is [107.57, 112.43]
(b)
n = 25
x-bar = 110
s = 7.1
% = 96
Standard Error, SE = s/n = 7.1/25 = 1.42
Degrees of freedom = n - 1 = 25 -1 = 24
t- score = 2.17154467
Width of the confidence interval = t * SE = 2.17154466985135 * 1.42 = 3.083593431
Lower Limit of the confidence interval = x-bar - width = 110 - 3.08359343118892 = 106.9164066
Upper Limit of the confidence interval = x-bar + width = 110 + 3.08359343118892 = 113.0835934
The 96% confidence interval is [106.92, 113.08]
(c)
N = (z * /E)^2 = (1.96 * 7.1/2.50^2 = 31
(d)
N = (z * /E)62 = (2.5758 * 7.1/3)^2 = 38.
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