Math Problem Statement
Solution
To solve this problem, we need to:
-
Find the regression equation by calculating the slope () and the y-intercept () of the linear regression line , where:
- : Overhead width
- : Weight
-
Predict the weight of a seal with an overhead width of 2.4 cm using the regression equation.
-
Determine if the prediction is valid, considering the significance level of 0.05 and the range of the data.
Step 1: Compute Regression Equation
The slope formula () is: The y-intercept () is: Where:
- : Number of data points
- : Mean of values
- : Mean of values
Step 2: Calculate Prediction
Substitute into the regression equation to find the predicted weight.
Step 3: Evaluate Validity of Prediction
The prediction is valid if (x = as np import pandas as pd
Data from the image
x = np.array([7.3, 7.4, 9.7, 9.3, 8.8, 8.5]) # Overhead Width (cm) y = np.array([139, 174, 268, 224, 224, 218]) # Weight (kg)
Number of data points
n = len(x)
Calculating means
mean_x = np.mean(x) mean_y = np.mean(y)
Calculating slope (m) and y-intercept (b)
m = (np.sum(x * y) - (np.sum(x) * np.sum(y)) / n) / (np.sum(x**2) - (np.sum(x)**2) / n) b = mean_y - m * mean_x
Regression equation
regression_equation = f"y = {round(b)} + {round(m, 1)}x"
Predicting weight for overhead width of 2.4 cm
predicted_weight = b + m * 2.4
Checking if prediction is valid (extrapolation check)
is_valid = 2.4 >= min(x) and 2.4 <= max(x)
regression_equation, predicted1. The regression equation is:
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The predicted weight for an overhead width of cm is approximately .
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The prediction is not valid, as falls outside the range of the given overhead widths ( to ). This is an example of extrapolation, which is unreliable.
Would you like further clarifications or step-by-step details?
Here are 5 questions to explore further:
- What does extrapolation mean, and why is it unreliable?
- How would increasing the range of data improve predictions?
- Can a seal realistically have a negative weight, and how should we interpret such results?
- What methods could be used to assess the significance of the regression model?
- Would transformations (e.g., logarithmic) of the data yield better predictions?
Tip: Always check whether the input values for prediction fall within
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Statistics
Prediction
Formulas
Slope formula: m = (Σxy - (Σx)(Σy)/n) / (Σx^2 - (Σx)^2/n)
Regression line: ŷ = b + mx
Mean formula: mean = Σ(values)/n
Theorems
Linear Regression Model
Extrapolation Validity
Suitable Grade Level
Grade 10-12 (Statistics/High School Math)
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