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A student wanted to know whether Centrum vitamins dissolve faster than the corresponding generic brand. The student used vinegar as a proxy for stomach acid and measured the time (in minutes) it took for a vitamin to completely dissolve. The results are shown next. $$ \begin{array}{lllll|lllll} &&{\text { Centrum }} &&&&&{\text { Generic Brand }} \\ \hline 2.73 & 3.07 & 3.30 & 3.35 & 3.12 & 6.57 & 6.23 & 6.17 & 7.17 & 5.77 \\ \hline 2.95 & 2.15 & 2.67 & 2.80 & 2.25 & 6.73 & 5.78 & 5.38 & 5.25 & 5.55 \\ \hline 2.60 & 2.57 & 4.02 & 3.02 & 2.15 & 5.50 & 6.50 & 7.42 & 6.47 & 6.30 \\ \hline 3.03 & 3.53 & 2.63 & 2.30 & 2.73 & 6.33 & 7.42 & 5.57 & 6.35 & 5.92 \\ \hline 3.92 & 2.38 & 3.25 & 4.00 & 3.63 & 5.35 & 7.25 & 7.58 & 6.50 & 4.97 \\ \hline 3.02 & 4.17 & 4.33 & 3.85 & 2.23 & 7.13 & 5.98 & 6.60 & 5.03 & 7.18 \\ \hline \end{array} $$ (a) Draw side-by-side boxplots for each vitamin type. (b) Which vitamin type has more dispersion? (c) Which vitamin type appears to dissolve faster?

Step by step solution

01

Organize Data

List all the dissolution times for Centrum and Generic Brand. Centrum: 2.73, 3.07, 3.30, 3.35, 3.12, 2.95, 2.15, 2.67, 2.80, 2.25, 2.60, 2.57, 4.02, 3.02, 2.15, 3.03, 3.53, 2.63, 2.30, 2.73, 3.92, 2.38, 3.25, 4.00, 3.63, 3.02, 4.17, 4.33, 3.85, 2.23. Generic Brand: 6.57, 6.23, 6.17, 7.17, 5.77, 6.73, 5.78, 5.38, 5.25, 5.55, 5.50, 6.50, 7.42, 6.47, 6.30, 6.33, 7.42, 5.57, 6.35, 5.92, 5.35, 7.25, 7.58, 6.50, 4.97, 7.13, 5.98, 6.60, 5.03, 7.18.

02

Draw Side-by-Side Boxplots

To create side-by-side boxplots, place the dissolution times of Centrum and Generic Brand on the same scale. Each boxplot should display the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. Use software or graph paper to plot these values for each type of vitamin.

03

Analyzing Dispersion

Compare the interquartile ranges (IQR) of the boxplots. The IQR is calculated by subtracting Q1 from Q3. The vitamin type with a larger IQR has more dispersion.

04

Determine Which Type Dissolves Faster

Compare the medians of the two boxplots. The vitamin type with the lower median dissolution time dissolves faster.

05

Interpret Findings

Based on the boxplots, IQR, and medians, conclude which vitamin dissolves faster and which has more dispersion. This should be clear from the visual representation and statistical measures.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Boxplots

Boxplots, also known as box-and-whisker plots, are a graphical representation of a dataset that displays the distribution of the data in terms of quartiles. They are especially useful for comparing distributions between different groups.
A boxplot consists of a few key components:

• The box: This represents the interquartile range (IQR) spanning from the first quartile (Q1) to the third quartile (Q3). The box contains the middle 50% of the data.
• The median: This is a line within the box that shows the median or the middle value of the dataset.
• Whiskers: These are lines extending from the box to the lowest and highest values within 1.5 times the IQR from Q1 and Q3, respectively.
• Outliers: These are data points that fall outside the whiskers and are plotted individually as points.

To create side-by-side boxplots, you align the boxplots of two datasets (such as Centrum and Generic Brand vitamin dissolution times) on a single scale. This visual comparison helps identify differences in data spread, central tendency, and outliers. In this exercise, drawing side-by-side boxplots provides an intuitive means to evaluate and compare the dissolution times of Centrum vitamins with the Generic Brand.

Interquartile Range

The Interquartile Range (IQR) is a measure of statistical dispersion, representing the range within which the central 50% of the data points lie.
It is calculated using the first quartile (Q1) and the third quartile (Q3). The formula for IQR is:
\[ IQR = Q3 - Q1 \]
The IQR is a robust measure because it is not affected by outliers or extreme values.

• First Quartile (Q1): The median of the lower half of the dataset; 25% of data points are below this value.
• Third Quartile (Q3): The median of the upper half of the dataset; 75% of data points are below this value.

In the context of this exercise, we calculate the IQR for both the Centrum and Generic Brand vitamin dissolution times to assess which has more variability or dispersion. A larger IQR indicates more spread in the dataset. When comparing the two vitamin types, the one with the larger IQR has more dispersion in dissolution times.

Median

The median is the middle value of a dataset when it is ordered from least to greatest.
It is a measure of central tendency that divides the dataset into two halves, with 50% of the data points falling below it and 50% above it.
The median is less affected by extreme values or outliers compared to the mean, making it a more accurate representation of the central location for skewed distributions.

• For an even number of observations, the median is the average of the two middle numbers.

To find out which vitamin dissolves faster, we compare the medians of the dissolution times for both Centrum and Generic Brand vitamins. The vitamin type with the lower median indicates a faster overall dissolution time. In this exercise, the median helps us quickly identify which vitamin brand generally dissolves faster in vinegar, simulating stomach acid conditions.