Question

observing that the proportion of blue candies in his bowl appeared to be less than that of the other colors, a student decided to compare the color distribution in randomly chosen bags of the candy to the theoretical distribution reported by the candy company’s consumer affairs. for his study, the student bought three bags of the colored candies from local stores and counted the number of each color. the average number of each color in the three bags (rounded to the nearest integer) was distributed as shown to the right. use this data to complete parts (a) through (c).
a. obtain a relative-frequency distribution.

color	frequency	relative frequency
yellow	119
red	103
orange	52
green	43
blue	43

(round to three decimal places as needed.)
b. draw a pie chart. choose the correct chart below.
○ a. ◉ b. ○ c. ○ d.
pie chart options with percentages: a: brown 30.7%, yellow 23.4%, red 19.9%, orange 9.6%, green 8.2%, blue 8.2%; b: brown 29.7%, yellow 20.1%, red 23.2%, orange 8.4%, green 8.4%, blue 10.2%; c: brown 29.1%, yellow 23.4%, red 20.3%, orange 10.7%, green 8.2%, blue 8.2%; d: brown 29.7%, yellow 23.2%, red 20.1%, orange 10.2%, green 8.4%, blue 8.4%

Explanation:

Part (a)

Step1: Calculate total frequency

First, we find the total number of candies by summing up all the frequencies.
$$152 + 119 + 103 + 52 + 43 + 43 = 512$$

Step2: Calculate relative frequency for each color

Relative frequency is calculated as $\frac{\text{Frequency of color}}{\text{Total frequency}}$.

• Brown: $\frac{152}{512} \approx 0.297$
• Yellow: $\frac{119}{512} \approx 0.232$ (Wait, earlier miscalculation. Let's recalculate: $119\div512\approx0.232$? Wait no, $119\div512 = 0.232421875\approx0.232$? Wait, no, let's do it properly. Wait, 152+119=271, +103=374, +52=426, +43=469, +43=512. Correct. So:
• Brown: $152/512 = 0.296875 \approx 0.297$
• Yellow: $119/512 \approx 0.2324 \approx 0.232$ (Wait, but let's check again. 119 divided by 512: 512*0.23 = 117.76, 119-117.76=1.24, 1.24/512≈0.0024, so 0.2324≈0.232)
• Red: $103/512 \approx 0.2012 \approx 0.201$
• Orange: $52/512 = 0.1015625 \approx 0.102$
• Green: $43/512 \approx 0.0840 \approx 0.084$
• Blue: $43/512 \approx 0.0840 \approx 0.084$

Wait, maybe I made a mistake earlier. Let's recalculate each:

• Brown: $152 \div 512 = 0.296875 \approx 0.297$
• Yellow: $119 \div 512 \approx 0.2324 \approx 0.232$ (Wait, but 119/512: 512*0.23 = 117.76, 119-117.76=1.24, 1.24/512≈0.0024, so 0.2324≈0.232)
• Red: $103 \div 512 \approx 0.2012 \approx 0.201$
• Orange: $52 \div 512 = 0.1015625 \approx 0.102$
• Green: $43 \div 512 \approx 0.0840 \approx 0.084$
• Blue: $43 \div 512 \approx 0.0840 \approx 0.084$

Wait, but let's check the total of relative frequencies: 0.297 + 0.232 + 0.201 + 0.102 + 0.084 + 0.084 = 0.297+0.232=0.529; +0.201=0.73; +0.102=0.832; +0.084=0.916; +0.084=1.0. Perfect, that adds up.

So the relative frequencies are:

• Brown: $\approx 0.297$
• Yellow: $\approx 0.232$
• Red: $\approx 0.201$
• Orange: $\approx 0.102$
• Green: $\approx 0.084$
• Blue: $\approx 0.084$

To determine the correct pie chart, we compare the relative frequencies we calculated with the percentages given in each option:

• Brown: ~29.7%, Yellow: ~23.2%, Red: ~20.1%, Orange: ~10.2%, Green: ~8.4%, Blue: ~8.4%.

Looking at the options:

• Option A: Brown 30.7%, Yellow 23.4%, Red 19.9%, Orange 9.6%, Green 8.2%, Blue 8.2% – Doesn't match.
• Option B: Brown 29.7%, Yellow 20.1%, Red 23.2%, Orange 8.4%, Green 8.4%, Blue 10.2% – Red and Yellow, Orange and Blue percentages swapped. Doesn't match.
• Option C: Brown 29.1%, Yellow 23.4%, Red 20.3%, Orange 10.7%, Green 8.2%, Blue 8.2% – Doesn't match.
• Option D: Brown 29.7%, Yellow 23.2%, Red 20.1%, Orange 10.2%, Green 8.4%, Blue 8.4% – Matches our calculated relative frequencies (converted to percentages: 29.7%, 23.2%, 20.1%, 10.2%, 8.4%, 8.4%).

Answer:

(for part a):

Color	Frequency	Relative Frequency
Yellow	119	0.232
Red	103	0.201
Orange	52	0.102
Green	43	0.084
Blue	43	0.084

Part (b)