Kevin and Mark took a random sample of 100 pieces of trail mix to determine the number of peanuts, raisins, and almonds in a container.
| | Kevin's Sample | Mark's Sample |
| -------- | -------------- | ------------- |
| Peanuts | 74 | 45 |
| Raisins | 10 | 17 |
| Almonds | 16 | 38 |
If each container of trail mix is known to have $40 \%$ peanuts, $40 \%$ almonds, and $20 \%$ raisins, which sample is a better representation of the actual population?
Answer (1)
Calculate the percentage of peanuts, raisins, and almonds in Kevin's and Mark's samples.
Calculate the absolute difference between the percentages in each sample and the actual percentages.
Sum the absolute differences for each sample to find the total error.
Compare the total errors; the sample with the smaller error is the better representation. Mark's sample is the better representation with a total error of 0.10 .
Explanation
Analyze the problem First, let's analyze the problem. We have the composition of trail mix samples taken by Kevin and Mark, and we know the actual composition of the trail mix. We need to determine which sample is a better representation of the actual composition. To do this, we will calculate the percentage of each component (peanuts, raisins, and almonds) in each sample and compare it to the actual percentages. The sample with the smallest total difference from the actual percentages will be considered the better representation.
Calculate percentages for Kevin's sample Next, we calculate the percentage of each component in Kevin's sample:
Peanuts: 100 74 = 74%
Raisins: 100 10 = 10%
Almonds: 100 16 = 16%
Calculate percentages for Mark's sample Now, we calculate the percentage of each component in Mark's sample:
Peanuts: 100 45 = 45%
Raisins: 100 17 = 17%
Almonds: 100 38 = 38%
Calculate absolute differences for Kevin's sample Next, we calculate the absolute differences between Kevin's sample percentages and the actual percentages:
Peanuts: ∣74% − 40%∣ = 34% = 0.34
Raisins: ∣10% − 20%∣ = 10% = 0.10
Almonds: ∣16% − 40%∣ = 24% = 0.24
Calculate absolute differences for Mark's sample Now, we calculate the absolute differences between Mark's sample percentages and the actual percentages:
Peanuts: ∣45% − 40%∣ = 5% = 0.05
Raisins: ∣17% − 20%∣ = 3% = 0.03
Almonds: ∣38% − 40%∣ = 2% = 0.02
Calculate total error for Kevin's sample Now, we calculate the total error for Kevin's sample by summing the absolute differences: 0.34 + 0.10 + 0.24 = 0.68
Calculate total error for Mark's sample Now, we calculate the total error for Mark's sample by summing the absolute differences: 0.05 + 0.03 + 0.02 = 0.10
Compare total errors and conclude Comparing the total errors, Kevin's total error is 0.68 and Mark's total error is 0.10 . Since Mark's total error is smaller than Kevin's total error, Mark's sample is a better representation of the actual population.
Examples
Imagine you are a quality control manager at a food packaging company. You need to ensure that each package of mixed nuts contains the correct proportions of different types of nuts. By taking samples and comparing their composition to the known proportions, you can determine whether the packaging process is accurate and consistent. This helps maintain product quality and customer satisfaction.
