document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165702>Question 1165702</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Compute and interpret the correlation coefficient for the following coded data:<BR>\n" );
document.write( "X (height) 15 18 11 19 20 9<BR>\n" );
document.write( "Y(weight) 17 21 24 10 25 16<BR>\n" );
document.write( "Fit a regression line Y on X and X on Y and coefficient of correlation? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852943 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852943\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "import pandas as pd<BR>\n" );
document.write( "from scipy.stats import pearsonr, linregress\r<BR>\n" );
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document.write( "# Data<BR>\n" );
document.write( "X = np.array([15, 18, 11, 19, 20, 9])<BR>\n" );
document.write( "Y = np.array([17, 21, 24, 10, 25, 16])<BR>\n" );
document.write( "n = len(X)\r<BR>\n" );
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document.write( "# 1. Compute sums for manual calculation (for verification and required components)<BR>\n" );
document.write( "sum_X = np.sum(X)<BR>\n" );
document.write( "sum_Y = np.sum(Y)<BR>\n" );
document.write( "sum_X_sq = np.sum(X**2)<BR>\n" );
document.write( "sum_Y_sq = np.sum(Y**2)<BR>\n" );
document.write( "sum_XY = np.sum(X * Y)\r<BR>\n" );
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document.write( "# Calculate the covariance numerator (Num) and variance denominators (Den_X, Den_Y)<BR>\n" );
document.write( "Num = n * sum_XY - sum_X * sum_Y<BR>\n" );
document.write( "Den_X_sq = n * sum_X_sq - sum_X**2<BR>\n" );
document.write( "Den_Y_sq = n * sum_Y_sq - sum_Y**2\r<BR>\n" );
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document.write( "# Correlation Coefficient (r)<BR>\n" );
document.write( "r = Num / np.sqrt(Den_X_sq * Den_Y_sq)\r<BR>\n" );
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document.write( "# 2. Regression Line Y on X: Y = b0 + b1*X<BR>\n" );
document.write( "b1_yx = Num / Den_X_sq<BR>\n" );
document.write( "mean_X = np.mean(X)<BR>\n" );
document.write( "mean_Y = np.mean(Y)<BR>\n" );
document.write( "b0_yx = mean_Y - b1_yx * mean_X\r<BR>\n" );
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document.write( "# 3. Regression Line X on Y: X = b'0 + b'1*Y<BR>\n" );
document.write( "b1_xy = Num / Den_Y_sq<BR>\n" );
document.write( "b0_xy = mean_X - b1_xy * mean_Y\r<BR>\n" );
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document.write( "print(f\"n = {n}\")<BR>\n" );
document.write( "print(f\"Sum X = {sum_X}, Sum Y = {sum_Y}\")<BR>\n" );
document.write( "print(f\"Sum X^2 = {sum_X_sq}, Sum Y^2 = {sum_Y_sq}\")<BR>\n" );
document.write( "print(f\"Sum XY = {sum_XY}\")<BR>\n" );
document.write( "print(\"-\" * 30)<BR>\n" );
document.write( "print(f\"Correlation Coefficient (r): {r}\")<BR>\n" );
document.write( "print(f\"Regression Y on X (b1): {b1_yx}\")<BR>\n" );
document.write( "print(f\"Regression Y on X (b0): {b0_yx}\")<BR>\n" );
document.write( "print(f\"Regression X on Y (b'1): {b1_xy}\")<BR>\n" );
document.write( "print(f\"Regression X on Y (b'0): {b0_xy}\")\r<BR>\n" );
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document.write( "# Verify r^2 = b1_yx * b1_xy<BR>\n" );
document.write( "r_sq_check = b1_yx * b1_xy<BR>\n" );
document.write( "print(f\"Check r^2 = b1_yx * b1_xy: {r_sq_check}\")<BR>\n" );
document.write( "print(f\"r^2: {r**2}\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "n = 6<BR>\n" );
document.write( "Sum X = 92, Sum Y = 113<BR>\n" );
document.write( "Sum X^2 = 1512, Sum Y^2 = 2287<BR>\n" );
document.write( "Sum XY = 1731<BR>\n" );
document.write( "------------------------------<BR>\n" );
document.write( "Correlation Coefficient (r): -0.013137168189865763<BR>\n" );
document.write( "Regression Y on X (b1): -0.01644736842105263<BR>\n" );
document.write( "Regression Y on X (b0): 19.085526315789473<BR>\n" );
document.write( "Regression X on Y (b'1): -0.01049317943336831<BR>\n" );
document.write( "Regression X on Y (b'0): 15.530954879328437<BR>\n" );
document.write( "Check r^2 = b1_yx * b1_xy: 0.00017258518804882087<BR>\n" );
document.write( "r^2: 0.0001725851880488209\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "The computations reveal a very weak, almost non-existent, negative linear relationship between the coded height (X) and weight (Y) data.\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## 1\. Coefficient of Correlation ($r$)\r<BR>\n" );
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document.write( "The **correlation coefficient ($r$)** is calculated as:<BR>\n" );
document.write( "$$r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n(\sum X^2) - (\sum X)^2][n(\sum Y^2) - (\sum Y)^2]}}$$\r<BR>\n" );
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document.write( "Using the computed summary statistics:\r<BR>\n" );
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document.write( "  * $\sum X = 92$<BR>\n" );
document.write( "  * $\sum Y = 113$<BR>\n" );
document.write( "  * $\sum XY = 1731$<BR>\n" );
document.write( "  * $n = 6$\r<BR>\n" );
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document.write( "$$r \approx \mathbf{-0.0131}$$\r<BR>\n" );
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document.write( "### Interpretation of $r$\r<BR>\n" );
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document.write( "The correlation coefficient $r \approx -0.0131$ indicates a **very weak, negligible negative linear relationship** between the two variables.\r<BR>\n" );
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document.write( "  * The value is very close to $0$, which suggests there is **virtually no linear association** between the coded height (X) and weight (Y) in this sample.<BR>\n" );
document.write( "  * The negative sign means that as X (height) slightly increases, Y (weight) tends to slightly decrease, but this trend is too small to be meaningful.\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## 2\. Regression Line Y on X\r<BR>\n" );
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document.write( "The regression line used to predict Y (weight) from X (height) is $\hat{Y} = b_0 + b_1X$.\r<BR>\n" );
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document.write( "### Slope ($b_1$)\r<BR>\n" );
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document.write( "$$b_1 = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum X^2) - (\sum X)^2} \approx \mathbf{-0.0164}$$\r<BR>\n" );
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document.write( "### Intercept ($b_0$)\r<BR>\n" );
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document.write( "$$b_0 = \bar{Y} - b_1\bar{X} \approx 18.8333 - (-0.0164)(15.3333) \approx \mathbf{19.0855}$$\r<BR>\n" );
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document.write( "The equation for the regression line Y on X is:<BR>\n" );
document.write( "$$\mathbf{\hat{Y} = 19.0855 - 0.0164X}$$\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "## 3\. Regression Line X on Y\r<BR>\n" );
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document.write( "The regression line used to predict X (height) from Y (weight) is $\hat{X} = b'_0 + b'_1Y$.\r<BR>\n" );
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document.write( "### Slope ($b'_1$)\r<BR>\n" );
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document.write( "$$b'_1 = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum Y^2) - (\sum Y)^2} \approx \mathbf{-0.0105}$$\r<BR>\n" );
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document.write( "### Intercept ($b'_0$)\r<BR>\n" );
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document.write( "$$b'_0 = \bar{X} - b'_1\bar{Y} \approx 15.3333 - (-0.0105)(18.8333) \approx \mathbf{15.5310}$$\r<BR>\n" );
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document.write( "The equation for the regression line X on Y is:<BR>\n" );
document.write( "$$\mathbf{\hat{X} = 15.5310 - 0.0105Y}$$ </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );