Data Regression

Grade 11 · statistics · 43 practice problems · read aloud

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Data Regression: Finding the Line of Best Fit

What is Regression? 🤔

Data regression finds the "line of best fit" through a scatter plot of data points. It helps you see the relationship between two variables and make predictions. For example, you could predict a student's test score based on their hours of study.

How to Perform Linear Regression

  1. Plot your data on a scatter plot.
  2. Determine the correlation (positive, negative, or none).
  3. Calculate the regression line using technology (calculator/software). The equation is always in the form: y = a + bx, where 'b' is the slope and 'a' is the y-intercept.
  4. Interpret the line. The slope tells you the rate of change. The y-intercept is the predicted value when x=0.
  5. Make predictions by substituting x-values into your equation.

Worked Examples

Example 1: Study Time vs. Test Score

Data shows: (1, 60), (2, 70), (3, 75), (4, 85)

  1. Plot points. They show a positive trend.
  2. Using a calculator, the regression line is: y = 52.5 + 8x
  3. Interpretation: Slope (b=8) means for each extra hour studied, the score increases by ~8 points. The y-intercept (a=52.5) is the predicted score with zero study.
  4. Predict the score for 5 hours: y = 52.5 + 8(5) = 92.5

Example 2: Car Age vs. Value

Data for a car: (0, $25,000), (2, $19,000), (5, $12,000)

  1. Plot shows a negative correlation.
  2. Regression line: y = 25400 - 2600x
  3. Interpretation: The car depreciates $2,600 per year.
  4. Predict value at 7 years: y = 25400 - 2600(7) = $7,200

Common Mistakes to Avoid 🚫

  • Extrapolation: Don't predict far outside your data range. A 20-year-old car isn't worth -$26,000!
  • Correlation ≠ Causation: Just because two things are related doesn't mean one causes the other.
  • Forgetting Units: Always include units when interpreting slope and intercept.
  • Rounded Values: Using overly rounded values from your calculator can lead to inaccurate predictions.

Tips & Tricks

  • Remember the regression line equation: y = a + bx
  • Use technology! Learn how to perform regression on your graphing calculator.
  • The line always passes through the point (x̄, ȳ), the means of x and y.
  • Check your answer's reasonableness. Does the prediction make sense?

How to Practice

Find real-world data sets online. Practice with your calculator until the steps are automatic. Create your own scatter plots from simple data (like daily temperature vs. ice cream sales) and find the line. Always ask: "What does this slope mean in this context?"

Practice problems

6 of the 43, worked through step by step — try them before opening the answer.

1 Aroha is designing a triangular garden plot with sides measuring 12 m, 16 m, and 20 m. She needs to calculate the area of the triangular plot to determine how much fertilizer to purchase. What is the area of Aroha's triangular garden in square meters?

Hint: Consider using Heron's formula which calculates the area of a triangle when you know all three side lengths. First find the semi-perimeter, then apply the formula.

Show the answer

Answer: 96

  1. Calculate the semi-perimeter (s) of the triangle s = (12 + 16 + 20) / 2 s = 48 / 2 s = 24 m
  2. Apply Heron's formula Area = sqrt[s(s - a)(s - b)(s - c)] Area = sqrt[24(24 - 12)(24 - 16)(24 - 20)] Area = sqrt[24 × 12 × 8 × 4]
  3. Calculate each multiplication 24 × 12 = 288 288 × 8 = 2304 2304 × 4 = 9216
  4. Take the square root Area = sqrt(9216) = 96 The area of Aroha's triangular garden is 96 square meters.

2 Sophia collected data on the relationship between study time and test scores for her math class. She found the linear regression equation to be y = 2.6x + 71, where x is study time in hours and y is the test score. If a student studies for 6 hours, what test score does the regression model predict?

Hint: In a linear regression equation, you substitute the given x-value into the equation to find the predicted y-value. For example, if the equation was y = 3x + 50 and someone studied 4 hours, you would calculate 3 times 4 plus 50.

Show the answer

Answer: 86.6

  1. The regression equation is y = 2.6x + 71
  2. Substitute x = 6 into the equation: y = 2.6(6) + 71
  3. Calculate 2.6 × 6 = 15.6
  4. Add 71 to 15.6: 15.6 + 71 = 86.6
  5. The predicted test score is 86.6

3 Matiu collected data on the number of hours students studied for their physics test and their test scores. He found the linear regression equation to be y = 6x + 58, where x is the study time in hours and y is the test score. If a student studied for 4 hours, what score would the regression equation predict?

Hint: Think about how to use a given equation to find an unknown value when you know one variable.

Show the answer

Answer: 82

  1. The regression equation is y = 6x + 58, where x is study time and y is test score.
  2. We're told the student studied for 4 hours, so x = 4.
  3. Substitute x = 4 into the equation: y = 6(4) + 58
  4. Calculate 6 × 4 = 24
  5. Add 24 + 58 = 82
  6. The predicted test score is 82.

4 Emma is analyzing the relationship between study time and test scores. She collected data from 10 students and found the linear regression equation to be y = 2.5x + 65, where x is study time in hours and y is the test score. If a student studies for 8 hours, what test score does the regression model predict?

Hint: Consider how to use the given equation to make a prediction by substituting the study time value into the equation.

Show the answer

Answer: 85

  1. The regression equation is y = 2.5x + 65
  2. Substitute x = 8 (study time in hours) into the equation
  3. y = 2.5(8) + 65
  4. Calculate 2.5 × 8 = 20
  5. Add 20 + 65 = 85
  6. The predicted test score is 85

5 Emma is analyzing the relationship between study time and test scores. She collected data from 11 students: (1, 65), (3, 71), (5, 77), (7, 83), (9, 89), (11, 95), (13, 101), (15, 107), (17, 113), (19, 119), (21, 125). The linear regression equation for this data is y = 3x + b. What is the value of the y-intercept b?

Hint: Consider how the y-intercept relates to the data points and the given slope. Think about what happens when x equals zero in the regression equation.

Show the answer

Answer: 62

  1. The regression equation is given as y = 3x + b with slope 3.
  2. To find b, we can use the fact that the regression line passes through the mean point (x̄, ȳ).
  3. Calculate the mean of x-values: (1+3+5+7+9+11+13+15+17+19+21)/11 = 121/11 = 11
  4. Calculate the mean of y-values: (65+71+77+83+89+95+101+107+113+119+125)/11 = 1045/11 = 95
  5. The mean point is (11, 95). Substitute into the regression equation: 95 = 3(11) + b
  6. Calculate: 95 = 33 + b
  7. Solve for b: b = 95 - 33 = 62

The answer is 62.

6 Kaia is analyzing the relationship between study time and test scores for her math class. She collected data from 11 students and found the linear regression equation to be y = 1.7x + 63, where x is study time in hours and y is the test score. If a student studied for 5 hours, what would be the predicted test score?

  1. A) 69.5
  2. B) 73.5
  3. C) 71.5
  4. D) 75.5

Hint: To make a prediction using a regression equation, substitute the given x-value into the equation and calculate the corresponding y-value. For example, if an equation was y = 2x + 50 and you wanted to predict for x = 3, you would calculate y = 2(3) + 50 = 56.

Show the answer

Answer: C) 71.5

  1. The regression equation is y = 1.7x + 63
  2. Substitute x = 5 into the equation: y = 1.7(5) + 63
  3. Calculate 1.7 × 5 = 8.5
  4. Add 8.5 + 63 = 71.5
  5. The predicted test score is 71.5

The correct answer is C.

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