Q.2.33 - Elementary Statistics

Question

M\&M Colors. Observing that the proportion of blue M\&Ms in his bowl of candy appeared to be less than that of the other colors, R. Fricker, Jr., decided to compare the color distribution in randomly chosen bags of M\&Ms to the theoretical distribution reported by M\&M/MARS consumer affairs. Fricker published his findings in the article "The Mysterious Case of the Blue M\&Ms" (Chance, Vol. 9(4), pp. 19-22). For his study, Fricker bought three bags of M\&Ms from local stores and counted the number of each color. The average number of each color in the three bags was distributed as shown in the following table.

a. obtain a relative-frequency distribution.

b. draw a pie chart.

c. construct a bar chart.

Step-by-Step Solution

Answer

a). The frequency distribution is,

$\begin{array}{|ccl|}\hline \text{ Color }& \text{ Frequency }& \text{ Relative Frequency }\\ \text{ Brown }& 152& \frac{152}{509}=0.2986\\ \text{ Yellow }& 114& \frac{114}{509}=0.2240\\ \text{ Red }& 106& \frac{106}{509}=0.2083\\ \text{ Orange }& 51& \frac{51}{509}=0.1002\\ \text{ Green }& 43& \frac{43}{509}=0.0845\\ \text{ Blue }& 43& \frac{43}{509}=0.0845\\ \text{ Total }& 509& 1\\ \hline\end{array}$

b). The pie-chart,

c). The bar-chart,

Part (a) Step 1: Given Information

The data is given as,

Part (a) Step 2: Explanation

The relative frequency can be calculated using the formula,

$\text{ Relative Frequency }=\frac{\text{ Frequency }}{\text{ NumberofObservations }}$

In the table below, the frequency distribution is presented.

$\begin{array}{|ccl|}\hline \text{ Color }& \text{ Frequency }& \text{ Relative Frequency }\\ \text{ Brown }& 152& \frac{152}{509}=0.2986\\ \text{ Yellow }& 114& \frac{114}{509}=0.2240\\ \text{ Red }& 106& \frac{106}{509}=0.2083\\ \text{ Orange }& 51& \frac{51}{509}=0.1002\\ \text{ Green }& 43& \frac{43}{509}=0.0845\\ \text{ Blue }& 43& \frac{43}{509}=0.0845\\ \text{ Total }& 509& 1\\ \hline\end{array}$

Part (b) Step 1: Given Information

The data is given as,

Part (b) Step 2: Explanation

MINITAB has been used to create the pie chart.

The portion for each class in a pie chart is calculated by multiplying ${360}^{\circ }$ by the appropriate relative frequencies.

The steps of drawing a pie chart are as follows:

- Select Graph > Pie Chart from the menu bar.

- Choose values for the chart from the table.

- Select Color from the Categorical variable.

- Under Summary variables, choose Frequency.

- Go to the Labels tab and then to the Slide Labels tab.

- Under Label pie pieces with, double-check Category name and Percentage.

- Select OK.

Figure 1 shows the pie chart for the supplied data set.

Part (c) Step 1: Given Information

The data is given as,

Part (c) Step 2: Explanation

MINITAB has been used to create the bar chart.

The steps to making a bar chart are as follows:

- Select Graph > Bar Chart from the drop-down menu.

- Select Values from a table from the Bars represent section.

- Click OK after selecting Simple.

- Add one column of FREQUENCY to Categorical variables.

- Select options for the chart

- Choose Show as Percentage.

- Select OK.

The supplied data set's bar chart is depicted in the picture below.