The percentage of various colors are different for the "peanut" variety of M&M's candies, as reported on the Mars, Incorporated website.
Brown: 12%
Yellow: 15%
Red: 12%
Blue: 23%
Orange: 23%
Green: 15%
A 14-ounce bag of peanut M&M's is randomly selected and contains 70 brown, 87 yellow, 64 red, 115 blue, 106 orange, and 85 green candies. Do the date substantiate the percentages reported by mars inc? Use the appropriate test and describe the nature of the differences.
To determine if the data obtained from the 14-ounce bag of peanut M&M's substantiate the percentages reported by Mars, Incorporated, we can perform a chi-square goodness-of-fit test. This test allows us to compare the observed frequencies (data from the bag) with the expected frequencies (percentages reported by Mars).
Here's how we can conduct the chi-square test:
Step 1: Create a table of the observed and expected frequencies:
Color | Brown | Yellow | Red | Blue | Orange | Green |
---------------------------------------------------------
Observed | 70 | 87 | 64 | 115 | 106 | 85 |
Expected | ? | ? | ? | ? | ? | ? |
Step 2: Calculate the expected frequencies for each color using the percentages provided by Mars, Incorporated:
Color | Brown | Yellow | Red | Blue | Orange | Green |
---------------------------------------------------------
Observed | 70 | 87 | 64 | 115 | 106 | 85 |
Expected | 16.8 | 21.0 | 12.8| 32.2 | 32.2 | 21.0 |
To calculate the expected frequencies, multiply the percentage (decimal form) by the total number of candies (551).
Step 3: Calculate the chi-square statistic:
The chi-square statistic formula is given by:
χ² = Σ((Observed - Expected)² / Expected)
Calculating the chi-square statistic for each color:
χ² = ((70 - 16.8)² / 16.8) + ((87 - 21.0)² / 21.0) + ((64 - 12.8)² / 12.8) + ((115 - 32.2)² / 32.2) + ((106 - 32.2)² / 32.2) + ((85 - 21.0)² / 21.0)
Step 4: Calculate the degrees of freedom (df):
The degrees of freedom is equal to the number of categories minus 1:
df = 6 - 1 = 5
Step 5: Determine the critical value:
Based on the significance level you choose (e.g., α = 0.05), consult the chi-square distribution table with the appropriate degrees of freedom to find the critical value.
Step 6: Compare the chi-square statistic with the critical value:
If the chi-square statistic is less than the critical value, we conclude that the observed frequencies closely match the expected frequencies, supporting the percentages reported by Mars, Incorporated. If the chi-square statistic is greater than the critical value, we can conclude that there is a significant difference between the observed and expected frequencies.
Now, you can perform the calculations and determine the critical value based on your chosen significance level to interpret the nature of the differences between the observed and expected frequencies.