An instructor gives regular 20-point quizzes and 100-point exams in a mathematics course. Average scores for six students, given as ordered pairs (x,y) where x is the average quiz score and y is the average test score, are (18, 87), (10, 55), (19, 96), (16, 79), (13, 76), and (15, 82). Use the calculator to find a function that will predict a student's average test grade as a function of this quiz average.

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To find a function that predicts a student's average test grade as a function of their quiz average, we can use the method of least squares regression. This involves finding the line of best fit that minimizes the sum of the squared differences between the predicted values and the actual values.

Let's use the given data points to calculate the regression line:

Data points:
(18, 87), (10, 55), (19, 96), (16, 79), (13, 76), (15, 82)

1) Calculate the average quiz score (x̄) and average test score (ȳ):
x̄ = (18 + 10 + 19 + 16 + 13 + 15) / 6 = 15.17 (rounded to two decimal places)
ȳ = (87 + 55 + 96 + 79 + 76 + 82) / 6 = 78.33 (rounded to two decimal places)

2) Calculate the deviations from the mean:
dx = x - x̄ and dy = y - ȳ for each data point

(18, 87): dx = 18 - 15.17 = 2.83, dy = 87 - 78.33 = 8.67
(10, 55): dx = 10 - 15.17 = -5.17, dy = 55 - 78.33 = -23.33
(19, 96): dx = 19 - 15.17 = 3.83, dy = 96 - 78.33 = 17.67
(16, 79): dx = 16 - 15.17 = 0.83, dy = 79 - 78.33 = 0.67
(13, 76): dx = 13 - 15.17 = -2.17, dy = 76 - 78.33 = -2.33
(15, 82): dx = 15 - 15.17 = -0.17, dy = 82 - 78.33 = 3.67

3) Calculate the sum of the products of the deviations:
Σ(dx * dy) = (2.83 * 8.67) + (-5.17 * -23.33) + (3.83 * 17.67) + (0.83 * 0.67) + (-2.17 * -2.33) + (-0.17 * 3.67)
Σ(dx * dy) = 24.5137 + 120.5721 + 67.5561 + 0.5561 + 5.0541 + -0.6239
Σ(dx * dy) = 217.6282 (rounded to four decimal places)

4) Calculate the sum of the squared deviations of the quiz scores:
Σ(dx^2) = (2.83^2) + (-5.17^2) + (3.83^2) + (0.83^2) + (-2.17^2) + (-0.17^2)
Σ(dx^2) = 8.0089 + 26.7289 + 14.6689 + 0.6889 + 4.7089 + 0.0289
Σ(dx^2) = 55.8324 (rounded to four decimal places)

5) Calculate the slope of the regression line:
slope (b) = Σ(dx * dy) / Σ(dx^2) = 217.6282 / 55.8324 = 3.8938 (rounded to four decimal places)

6) Calculate the y-intercept of the regression line:
y-intercept (a) = ȳ - (slope * x̄) = 78.33 - (3.8938 * 15.17) = 19.4704 (rounded to four decimal places)

7) Write the equation of the regression line:
y = ax + b
y = 19.4704 + (3.8938 * x)

Therefore, the function that predicts a student's average test grade as a function of their quiz average is:
Test Score = 19.4704 + (3.8938 * Quiz Score)

You can use this equation to predict a student's average test score based on their quiz average.