Karen took three samples from a population of students to measure their average study time.
| Sample | Predicted Mean | Actual Mean |
| -------- | -------------- | ----------- |
| Sample 1 | 34 | 28 |
| Sample 2 | 15 | 28 |
| Sample 3 | 29 | 28 |
Compute the variation of each predicted population mean from the sample means in the table.
Sample 1:
Sample 2:
Sample 3:
Answer (2)
Calculate the variation for Sample 1 by finding the absolute difference between the predicted mean (34) and the actual mean (28), resulting in ∣34 − 28∣ = 6 .
Calculate the variation for Sample 2 by finding the absolute difference between the predicted mean (15) and the actual mean (28), resulting in ∣15 − 28∣ = 13 .
Calculate the variation for Sample 3 by finding the absolute difference between the predicted mean (29) and the actual mean (28), resulting in ∣29 − 28∣ = 1 .
The variations for the three samples are 6, 13, and 1, respectively, which means the final answer is:
6 , 13 , 1
Explanation
Understanding the Problem We are given a table with predicted and actual means for three samples. The problem asks us to compute the variation of each predicted population mean from the actual sample means. The variation is calculated as the absolute difference between the predicted mean and the actual mean for each sample.
Calculating Variation for Sample 1 For Sample 1, the predicted mean is 34 and the actual mean is 28. The variation is calculated as follows: ∣34 − 28∣ = ∣6∣ = 6 So, the variation for Sample 1 is 6.
Calculating Variation for Sample 2 For Sample 2, the predicted mean is 15 and the actual mean is 28. The variation is calculated as follows: ∣15 − 28∣ = ∣ − 13∣ = 13 So, the variation for Sample 2 is 13.
Calculating Variation for Sample 3 For Sample 3, the predicted mean is 29 and the actual mean is 28. The variation is calculated as follows: ∣29 − 28∣ = ∣1∣ = 1 So, the variation for Sample 3 is 1.
Final Answer The variations for the three samples are: Sample 1: 6 Sample 2: 13 Sample 3: 1
Examples
Understanding variation is crucial in many real-world scenarios. For instance, in weather forecasting, there's often a predicted temperature and an actual temperature. The variation between these two values helps meteorologists refine their models and improve future predictions. Similarly, in manufacturing, the predicted weight of a product might differ slightly from its actual weight. Analyzing this variation helps engineers optimize the production process and ensure quality control. By calculating the absolute difference, we focus on the magnitude of the difference, regardless of whether the prediction was an overestimate or an underestimate, providing a clear measure of accuracy.
The variations for the three samples are 6 for Sample 1, 13 for Sample 2, and 1 for Sample 3, calculated as the absolute differences between the predicted and actual means.
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