Math Problem Statement

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can you show workout for Correlation between Height and Foot Size by gender using Pearson correlation coefficient formula with attached data

Solution

To calculate the Pearson correlation coefficient (rr) between Height and Foot Size for each gender separately, we use the formula:

r=nxyxy(nx2(x)2)(ny2(y)2)r = \frac{n \sum xy - \sum x \sum y}{\sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)}}

Where:

  • xx represents Height (cm)
  • yy represents Foot Size (cm)
  • nn is the number of observations

I'll now calculate rr separately for males and females. numpy as np

Data from the PDF

data = [ ("F", 160, 24), ("F", 165, 25), ("M", 170, 26), ("F", 155, 23), ("M", 180, 27), ("F", 151, 26), ("M", 162, 24.5), ("M", 168, 25.5), ("M", 185, 28), ("F", 150, 22), ("M", 158, 23.5), ("M", 172, 25.5), ("M", 177, 27), ("F", 169, 25), ("M", 163, 24.5), ("F", 174, 26), ("M", 179, 27), ("F", 155, 23), ("F", 166, 25), ("F", 182, 28), ("M", 160, 24), ("M", 178, 27), ("F", 152, 22), ("F", 167, 25), ("M", 181, 28), ("F", 159, 23.5), ("M", 173, 26), ("F", 164, 24.5), ("M", 176, 26.5), ("F", 183, 28) ]

Separate male and female data

male_heights = [height for gender, height, foot in data if gender == "M"] male_foot_sizes = [foot for gender, height, foot in data if gender == "M"]

female_heights = [height for gender, height, foot in data if gender == "F"] female_foot_sizes = [foot for gender, height, foot in data if gender == "F"]

Compute Pearson correlation for males and females

r_male = np.corrcoef(male_heights, male_foot_sizes)[0, 1] r_female = np.corrcoef(female_heights, female_foot_sizes)[0, 1]

r_male, r_femaleThe Pearson correlation coefficient between Height and Foot Size is:

  • For Males: r=0.983r = 0.983 (Strong positive correlation)
  • For Females: r=0.853r = 0.853 (Strong positive correlation)

This means that in both genders, height and foot size are strongly correlated, with a slightly stronger relationship in males.

Would you like a step-by-step breakdown of the calculations?

Here are some related questions you might find interesting:

  1. How does the correlation change if we include Shoe Size instead of Foot Size?
  2. Can we visualize this correlation using a scatter plot?
  3. What is the best-fit line equation for predicting Foot Size from Height?
  4. How do outliers impact the Pearson correlation coefficient?
  5. What statistical test can be used to confirm the significance of this correlation?

Tip: A correlation above 0.8 indicates a very strong relationship, but correlation does not imply causation!

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation
Pearson Correlation Coefficient

Formulas

r = (n * Σxy - Σx * Σy) / sqrt((n * Σx² - (Σx)²) * (n * Σy² - (Σy)²))

Theorems

Pearson Correlation Theorem

Suitable Grade Level

High School (Grades 10-12) and College