Math Problem Statement

Given the table of data points for x and y, determine the regression equation y = a + bx, and predict the value of y for x = 6. Additionally, identify the matching scatterplot from the options.

Solution

To assist you, I'll extract the mathematical content from the image and guide you step-by-step.


1. Observations:

From the table in the image:

  • x={2,6,9,13,20}x = \{2, 6, 9, 13, 20\}
  • y={7,18,9,25,23}y = \{7, 18, 9, 25, 23\}

2. Subtasks:

(a) Identify the scatterplot that represents the data.
I'll compare the data points with the scatterplots labeled A, B, C, and D.

(b) Develop the regression equation y=a+bxy = a + bx to 1 decimal place.
We will compute:

  1. b=(xixˉ)(yiyˉ)(xixˉ)2b = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
  2. a=yˉbxˉa = \bar{y} - b\bar{x}

(c) Use the regression equation to predict yy when x=6x = 6.


Let me perform the calculations now. numpy as np

Data points

x = np.array([2, 6, 9, 13, 20]) y = np.array([7, 18, 9, 25, 23])

Number of points

n = len(x)

Mean of x and y

x_mean = np.mean(x) y_mean = np.mean(y)

Calculating the slope (b)

b_numerator = np.sum((x - x_mean) * (y - y_mean)) b_denominator = np.sum((x - x_mean) ** 2) b = b_numerator / b_denominator

Calculating the intercept (a)

a = y_mean - b * x_mean

Regression equation: y = a + bx (rounded to 1 decimal place)

a_rounded = round(a, 1) b_rounded = round(b, 1)

Prediction for x = 6

x_pred = 6 y_pred = a + b * x_pred y_pred_rounded = round(y_pred, 1)

a_rounded, b_rounded, y_pred_rounded### Results:

  1. Regression Equation:
    The estimated regression equation is: y=7.6+0.9xy = 7.6 + 0.9x

  2. Prediction for x=6x = 6:
    Using the regression equation, the predicted value of yy when x=6x = 6 is: y=12.9y = 12.9


Scatterplot:

Now I'll analyze the scatterplots to identify the one that matches the data points. Let me process this visually and confirm.### Matching Scatterplot:

By comparing the table of points with the scatterplots:

  • The data points (x,y)=(2,7),(6,18),(9,9),(13,25),(20,23)(x, y) = (2, 7), (6, 18), (9, 9), (13, 25), (20, 23) align with Scatterplot C.
    Scatterplot C accurately represents the data.

Summary:

  1. Scatterplot: C.
  2. Regression Equation: y=7.6+0.9xy = 7.6 + 0.9x.
  3. Prediction: y=12.9y = 12.9 when x=6x = 6.

Would you like detailed steps for the regression calculations or have additional questions?
Here are related questions to explore:

  1. How do we derive the slope formula in regression?
  2. Why do outliers affect regression results?
  3. Can you explain the interpretation of slope and intercept in this context?
  4. What methods can validate the regression model?
  5. What are limitations of linear regression for this dataset?

Tip: Always check scatterplots for clear trends before applying regression analysis.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistics
Scatterplots
Predictions

Formulas

b = Σ[(x_i - x̄)(y_i - ȳ)] / Σ[(x_i - x̄)^2]
a = ȳ - b * x̄

Theorems

Least Squares Regression
Prediction from Regression Equation

Suitable Grade Level

Grades 9-12