Question 1179680: During one term in Math 310, Min Lee took seven tests, the last of which carried twice the weight of
each of the others when averages were computed. Min’s test-score vector for the term was
[84,78,91,80,72,88,83]. Show that Min’s final average, a weighted average, can be calculated as a dot
product of this vector with another seven-component vector. How can the teacher obtain a class list of
test averages by multiplying two matrices?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's break down how to represent Min Lee's weighted average as a dot product and how to calculate class averages using matrix multiplication.
**1. Min Lee's Weighted Average as a Dot Product:**
* **Min's Test Scores:** [84, 78, 91, 80, 72, 88, 83]
* **Weights:** The first six tests have a weight of 1, and the last test has a weight of 2. So the weights are [1, 1, 1, 1, 1, 1, 2].
* **Total Weight:** 1 + 1 + 1 + 1 + 1 + 1 + 2 = 8
To calculate Min's weighted average as a dot product, we need to normalize the weights by dividing each weight by the total weight:
* **Normalized Weights:** [1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 2/8] = [0.125, 0.125, 0.125, 0.125, 0.125, 0.125, 0.25]
Now, we can calculate the dot product of Min's test scores and the normalized weights:
* Weighted Average = (84 * 0.125) + (78 * 0.125) + (91 * 0.125) + (80 * 0.125) + (72 * 0.125) + (88 * 0.125) + (83 * 0.25)
* Weighted Average = 10.5 + 9.75 + 11.375 + 10 + 9 + 11 + 20.75
* Weighted Average = 82.375
This is the same as:
```
import numpy as np
# Min's test scores
min_scores = np.array([84, 78, 91, 80, 72, 88, 83])
# Weights for each test
weights = np.array([1, 1, 1, 1, 1, 1, 2])
# Calculate Min's final average using dot product
min_average = np.dot(min_scores, weights) / sum(weights)
print(f"Min's final average: {min_average}")
```
**2. Class List of Test Averages Using Matrix Multiplication:**
Let's say the teacher has a class of 'm' students and the same 'n' number of tests. We can represent the test scores in a matrix called 'Scores', where:
* Scores is an m x n matrix (m rows, n columns).
* Each row represents a student's test scores.
* Each column represents the scores for a particular test.
We can also represent the normalized weights in a column vector called 'Weights', where:
* Weights is an n x 1 matrix (n rows, 1 column).
* Each element represents the normalized weight for a test.
To get the class list of test averages, we can multiply the 'Scores' matrix by the 'Weights' vector:
* Averages = Scores * Weights
The result will be an m x 1 matrix (a column vector) where each element represents a student's weighted average.
Here's how it would look in Python using NumPy:
```python
import numpy as np
# Example: 3 students, 7 tests
scores = np.array([
[84, 78, 91, 80, 72, 88, 83], # Min Lee's scores
[90, 85, 78, 92, 88, 95, 90], # Another student's scores
[75, 82, 70, 78, 80, 75, 81] # Another student's scores
])
weights = np.array([1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 2/8])
class_averages = np.dot(scores, weights)
print("Class Averages:", class_averages)
```
In essence, matrix multiplication allows the teacher to efficiently calculate the weighted averages for all students in the class with a single operation.
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