Question

Fiber content (in grams per serving) and sugar content (in grams per serving) for 18 high-fiber cereals (www .consumerreports.com) are shown. Fiber Content \(\begin{array}{rrrrrrr}7 & 10 & 10 & 7 & 8 & 7 & 12 \\ 12 & 8 & 13 & 10 & 8 & 12 & 7 \\ 14 & 7 & 8 & 8 & & & \end{array}\) Sugar Content \(\begin{array}{rrrrrrr}11 & 6 & 14 & 13 & 0 & 18 & 9 \\ 10 & 19 & 6 & 10 & 17 & 10 & 10 \\ 0 & 9 & 5 & 11 & & & \end{array}\) a. Find the median, quartiles, and interquartile range for the fiber content data set. b. Find the median, quartiles, and interquartile range for the sugar content data set. c. Are there any outliers in the sugar content data set? d. Explain why the minimum value and the lower quartile are equal for the fiber content data set. e. Construct a comparative boxplot and use it to comment on the differences and similarities in the fiber and sugar distributions.

Short answer

Solution will include calculated measures of central tendency, spread and identification of any outliers as per these calculations. Especially for the fiber content dataset it seems likely that its minimum would be equal to first quartile, because the dataset contains numerous similar values that are lower than the rest of the data. Additionally, a comparative boxplot will be created and analysed, showcasing the difference in dispersion and median between the two datasets.

Step-by-step solution

Arrange The Data

Arrange data for Fiber content and Sugar content in ascending order.

Calculate Median

Find the middle value for Fiber and Sugar content datasets. If number of observations is odd, middle value is the median; if it is even, median is the average of two middle numbers.

Find Quartiles

\(Q_1\) is calculated as median of the first half of the data and \(Q_3\) as median of the second half of the data. The boundaries for halves do not include the previously found median if count of numbers in the dataset is odd.

Establish Interquartile Range

Interquartile range (IQR) is the found by subtracting \(Q_1\) from \(Q_3\) for both datasets.

Outlier Detection

For the sugar content, detect outliers by identifying numbers that fall below \(Q_1−1.5\times IQR\) or above \(Q_3+1.5\times IQR\).

Boxplot Analysis

Create graphical representations (boxplots) by plotting \(Q_1\), median, \(Q_3\), and any outliers for both datasets. Analyze the similarity and difference between these boxplots.

Analyze Min Value And Lower Quartile

Explain why the minimum value and the lower quartile are equal for the Fiber content data set. This happens when 25% of the data values are the same as the minimum value.