The caffeine content of 8-ounce 33 milligrams (mg). A randomly selected 8-ounce can containing 35 mg of caffeine is 1.2 standard deviations above the mean. Approximately what percent of 8-ounce cans of the cola have a caffeine content greater than 35 mg? A) )1% (B) 8% (C) 12% (D) 16%
Mathematics · College · Thu Feb 04 2021
Answered on
To answer this question, we need to understand how the standard deviation works within a normal distribution. When a data point is a certain number of standard deviations away from the mean, we can use the properties of the normal distribution to determine what percentage of values lie beyond that point.
In a normal distribution: - Approximately 68% of the data falls within one standard deviation of the mean. - About 95% of the data falls within two standard deviations of the mean. - Around 99.7% of the data falls within three standard deviations of the mean.
A can of cola that has 35 mg of caffeine is 1.2 standard deviations above the mean. In a normal distribution, approximately 84.13% of data falls below one standard deviation above the mean, because 50% of the values are below the mean, and an additional 34.13% is between the mean and one standard deviation above it.
Since we're looking at 1.2 standard deviations above the mean, less than 84.13% of the cans will have caffeine contents below 35 mg. This means that more than (100% - 84.13% = 15.87%) of the cans will have contents greater than 35 mg. Now, let's find the exact percentage using the z-score table or normal distribution table:
A z-score of 1.2 corresponds roughly to the 88.49th percentile (this value can be found in a standard z-score table or calculated using statistical software). This means that approximately 88.49% of values are below a value that is 1.2 standard deviations above the mean.
Therefore, the percentage of cans with a caffeine content greater than 35 mg is roughly 100% - 88.49% = 11.51%.
Looking at the options, option C (12%) is the closest to the calculated value, so this would be the estimated percentage of 8-ounce cans of the cola that have a caffeine content greater than 35 mg.