Q 46. - The Practice of Statistics for AP

Question

Watch that caffeine! The U.S. Food and Drug Administration (USFDA) limits the amount of caffeine in a $12$-ounce can of carbonated beverage to$72$ milligrams. That translates to a maximum of$48$ milligrams of caffeine per $8$-ounce serving. Data on the caffeine content of popular soft drinks (in milligrams per $8$-ounce serving) are displayed in the stem plot below.

(a) Why did we split stems?

(b) Give an appropriate key for this graph.

(c) Describe the shape, center, and spread of the distribution. Compare the caffeine content of these drinks with the USFDA’s limit.

Step-by-Step Solution

Answer

Part (a) To get a better picture of the distribution, the stems are split.

Part (b) $2|4$ means that $24$ milligrams of caffeine content in an$8$-ounce can of carbonated beverage

Part (c) 

• Shape: skewed to the right
• Center: $28$mg of caffeine per$8$-ounce serving.
• Spread: $15$ mg to $47$mg of caffeine per $8$-ounce serving.
• Outliers: none

Part (a) Step 1: Given information

The stemplot is,

Part (a) Step 2: Concept

Shape: The peak of the dot plot is at  This means that during that season, the team's most common number of goals scored in games (known as the mode) was  The United States women's soccer team scored between one and five goals in the majority of its games. The distribution, however, has a large tail to the right. 

Centre: can be described by selecting a value that divides the observations in half, with the bigger half taking greater values and the smaller half taking lower values. 

The spread of distribution indicates the amount of variability in the data. Giving the smallest and greatest values is one approach to describe the variability.

Part (a) Step 3: Explanation

To get a better picture of the distribution, the stem has been dissected. It has two distinct peaks at $2$ and $3$ The dispersion is better viewed or shaped when the stem is split.

Part (b) Step 1: Explanation

$2|4$ indicates that an $8$-ounce can of carbonated beverage contains $24$ mg of caffeine.

Part (c) Step 1: Explanation

Because most of the data values are near the top of the stem plot, the distribution is skewed to the right.

The median, or the middle (${19}^{th}$ data value), can be used to signify the center of the distribution. The stem plot's 19th data value is $2|8$, indicating that the center contains $28$ mg of caffeine per $8$-ounce serving.

Spread: $15$ mg is the smallest value and $47$ mg is the biggest. As a result, per $8$-ounce serving, caffeine levels will range from $15$ to $47$ mg.

There don't appear to be any outliers because the stem plot doesn't have any gaps (there are no stems without leaves).