Question

The paper "Caffeinated Energy Drinks-A Growing Problem" (Drug and Alcohol Dependence [2009]: 1-10) reported caffeine per ounce for 8 top-selling energy drinks and for 11 highcaffeine energy drinks: Top Selling Energy Drinks $$ \begin{array}{llll} 9.6 & 10.0 & 10.0 & 9.0 \end{array} $$ 9.5 High-Caffeine Energy Drinks \(21.0 \quad 2\) 25.0 15.0 ! \(21 .\) \(\begin{array}{ll}5 & 35 .\end{array}\) 30.0 \(31.3 \quad 3\) \(33.3 \quad 11.9 \quad 16.3\) The mean caffeine per ounce is clearly higher for the highcaffeine energy drinks, but which of the two groups of energy drinks is the most variable? Justify your choice.

Short answer

After performing the calculations above, compare the variance values obtained to determine which group of energy drinks shows more variability. This will be the group with the larger variance value.

Step-by-step solution

Organize the Data

The first thing you need to do is to organize the data which is given as follows: \nTop Selling Energy Drinks (in mg/oz): 9.6, 10.0, 10.0, 9.0, 9.5; and High-Caffeine Energy Drinks (in mg/oz): 21.0, 25.0, 15.0, 21.5, 35.0, 30.0, 31.3, 33.3, 11.9, 16.3.

Find the Mean

Next, calculate the mean of both datasets. The mean is calculated as the sum of all the numbers in the set divided by the count of numbers in the set. For the Top Selling Energy Drinks: Mean = (9.6 + 10.0 + 10.0 + 9.0 + 9.5) / 5 = 9.62. For the High-Caffeine Energy Drinks: Mean = (21.0 + 25.0 + 15.0 + 21.5 + 35.0 + 30.0 + 31.3 + 33.3 + 11.9 + 16.3) / 10 = 23.03.

Calculate the Variance

Next, calculate the variance for each of the datasets. The variance is calculated by subtract all the mean from each number in the set, squaring the result, and then averaging those square differences. For the Top Selling Energy Drinks, you have \[\Var_{Sold} = \frac {1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\], and for the High-Caffeine Energy Drinks, you have \[\Var_{High} = \frac {1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\]. Calculate these values.

Compare Variance

Compare the variance values from the Top Selling Energy Drinks and High-Caffeine Energy Drinks data sets. The group with the larger variance is the more variable.

Key concepts

Data Organization

Organizing data is a crucial step in statistical analysis. It's about arranging data in a systematic way that makes it easier to work with and understand. For instance, in the exercise involving energy drinks, the data was presented as caffeine content per ounce for two groups of drinks. These groups were: Top Selling Energy Drinks, and High-Caffeine Energy Drinks.

To analyze such data, it is important to list all values for each group clearly. This helps in ensuring that no data has been missed and makes further calculations, such as mean and variance, more straightforward. A well-organized dataset also reduces the likelihood of errors when performing statistical computations.

Mean Calculation

The mean, often referred to as the average, is a measure of central tendency that sums up the essence of a data set with a single number. Calculating the mean involves adding up all data points and dividing the result by the total number of points. This gives us a representative value for the data. In the exercise, the mean caffeine per ounce is calculated separately for the two groups of energy drinks.

The mean allows us to compare different sets and identify which one tends to have greater or lesser values. It's a fundamental step in understanding the general behavior of the dataset before we dive deeper into variability or dispersion measures like variance.

Variance Calculation

Variance is a statistical measure that tells us how spread out or dispersed the values in a dataset are. It's calculated by taking the average of the squared differences from the mean. This process involves subtracting the mean from each number to find the deviation from the mean, squaring that deviation (to get rid of negative values and give more weight to larger deviations), and then finding the average of these squared deviations.

In the given exercise, we calculate variance for both groups of energy drinks, using the formula \[\Var_{x} = \frac {1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\], where \( \bar{x} \) is the mean, \( x_i \) represents each value in the dataset, and \( n \) is the total number of values. The resulting variance gives us insight into how much the caffeine content varies within each group of energy drinks.

Comparing Variance

When comparing variance between two or more groups, it helps us understand which group has greater inconsistency in its data points. A higher variance signifies that the data points are more spread out from the mean, indicating greater variability among the values. Conversely, a lower variance shows that data points are closer to the mean, suggesting less variability.

In our exercise, comparing the variance of the caffeine content between the Top Selling Energy Drinks and High-Caffeine Energy Drinks will tell us which group has a more variable amount of caffeine per ounce. This comparison can be crucial for quality control, customer insights, or even health recommendations. By understanding the variability within datasets, we can make more informed decisions based on the consistency of the data.