Math Problem Statement

x y 3 90.8 4 91.2 6 82.5 8 75.8 9 62.8 17 59.1 19 60.5 Given that the linear relationship between x and y is statistically significant, find the linear regression line in the form ŷ =ax+by^=ax+b for this relationship. Round a and b to the nearest thousandth find ŷ=________

Solution

import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression

Data provided in the question

data = { "x": [3, 4, 6, 8, 9, 17, 19], "y": [90.8, 91.2, 82.5, 75.8, 62.8, 59.1, 60.5] }

Convert data to a DataFrame

df = pd.DataFrame(data)

Prepare the variables for regression

X = np.array(df['x']).reshape(-1, 1) y = np.array(df['y'])

Perform linear regression

model = LinearRegression() model.fit(X, y)

Get the coefficients

a = model.coef_[0] b = model.intercept_

Round to the nearest thousandth

a_rounded = round(a, 3) b_rounded = round(b, 3)

a_rounded, b_rounded(-2.012, 93.645)(-2.012, 93.645)successThe linear regression line for the relationship between $x$ and $y$ is:

$\hat{y} = -2.012x + 93.645$

Let me know if you want further details or have any questions!

Tip:

Always visualize your data with the regression line to ensure the model fits well!

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistics
Algebra

Formulas

ŷ = ax + b
a = Cov(x, y) / Var(x)
b = ȳ - a * x̄

Theorems

Least Squares Method

Suitable Grade Level

Grades 11-12

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