document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162465>Question 1162465</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Raw material used in the production of synthetic fiber is stored in a place which has no humidity control. Measurements of the relative humidity in the storage place and the moisture content of a sample of the raw material (both in percentages) on 12 days yielded the following results: <BR>\n" );
document.write( " Humidity x         42  C5 50 4C 48 62 C1 C6 44 C9 55 48             <BR>\n" );
document.write( "Moixture content y  12  8  14  9  11  16  7  9  12  10  1C  11<BR>\n" );
document.write( "(i) Make a scatter plot to verify that it is reasonable to assume that the regression of y on x is linear (ii) Fit a straight line by the method of least squares, (iii) Estimate the moisture content when humidity is 45%. (iv) Determine the correlation coefficient.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854403 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</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=854403\">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 matplotlib.pyplot as plt<BR>\n" );
document.write( "from scipy.stats import linregress\r<BR>\n" );
document.write( "\n" );
document.write( "# Data with C interpreted as 3<BR>\n" );
document.write( "x = np.array([42, 35, 50, 43, 48, 62, 31, 36, 44, 39, 55, 48])<BR>\n" );
document.write( "y = np.array([12, 8, 14, 9, 11, 16, 7, 9, 12, 10, 13, 11])\r<BR>\n" );
document.write( "\n" );
document.write( "# (i) Scatter plot<BR>\n" );
document.write( "plt.scatter(x, y, color='blue', label='Data Points')<BR>\n" );
document.write( "plt.xlabel('Relative Humidity x (%)')<BR>\n" );
document.write( "plt.ylabel('Moisture Content y (%)')<BR>\n" );
document.write( "plt.title('Scatter Plot of Humidity vs. Moisture Content')\r<BR>\n" );
document.write( "\n" );
document.write( "# (ii) Least Squares Fit<BR>\n" );
document.write( "slope, intercept, r_value, p_value, std_err = linregress(x, y)<BR>\n" );
document.write( "line = slope * x + intercept<BR>\n" );
document.write( "plt.plot(x, line, color='red', label=f'Fit: y={intercept:.2f} + {slope:.2f}x')<BR>\n" );
document.write( "plt.legend()<BR>\n" );
document.write( "plt.grid(True)<BR>\n" );
document.write( "plt.savefig('scatter_plot.png')\r<BR>\n" );
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document.write( "# (iii) Estimate at x = 45%<BR>\n" );
document.write( "x_est = 45<BR>\n" );
document.write( "y_est = intercept + slope * x_est\r<BR>\n" );
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document.write( "# (iv) Correlation coefficient<BR>\n" );
document.write( "correlation = r_value\r<BR>\n" );
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document.write( "print(f\"Slope (b): {slope}\")<BR>\n" );
document.write( "print(f\"Intercept (a): {intercept}\")<BR>\n" );
document.write( "print(f\"Estimate for x=45: {y_est}\")<BR>\n" );
document.write( "print(f\"Correlation coefficient (r): {correlation}\")\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Slope (b): 0.26903206940247587<BR>\n" );
document.write( "Intercept (a): -0.9495077492933035<BR>\n" );
document.write( "Estimate for x=45: 11.156935373818111<BR>\n" );
document.write( "Correlation coefficient (r): 0.9144291969589217\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
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document.write( "[image-tag: code-generated-image-0-1776522998299002780]\r<BR>\n" );
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document.write( "To analyze the relationship between relative humidity ($x$) and moisture content ($y$), we first interpreted the characters in the data (where 'C' appeared to represent '3'). \r<BR>\n" );
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document.write( "**The clean dataset used for analysis:**<BR>\n" );
document.write( "* **Humidity ($x$):** 42, 35, 50, 43, 48, 62, 31, 36, 44, 39, 55, 48<BR>\n" );
document.write( "* **Moisture ($y$):** 12, 8, 14, 9, 11, 16, 7, 9, 12, 10, 13, 11\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
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document.write( "### (i) Scatter Plot<BR>\n" );
document.write( "The scatter plot shows a strong, positive linear trend. As relative humidity increases, the moisture content of the raw material also tends to increase. The points cluster closely around a straight line, making the assumption of a linear regression model very reasonable.\r<BR>\n" );
document.write( "\n" );
document.write( "### (ii) Least Squares Regression Line<BR>\n" );
document.write( "Using the method of least squares, we calculated the regression parameters:<BR>\n" );
document.write( "* **Slope ($b$):** $0.2690$<BR>\n" );
document.write( "* **Intercept ($a$):** $-0.9495$\r<BR>\n" );
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document.write( "The equation of the fitted straight line is:<BR>\n" );
document.write( "$$\hat{y} = -0.9495 + 0.2690x$$\r<BR>\n" );
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document.write( "### (iii) Estimate for 45% Humidity<BR>\n" );
document.write( "To estimate the moisture content when the relative humidity is 45%, we substitute $x = 45$ into our regression equation:<BR>\n" );
document.write( "$$\hat{y} = -0.9495 + 0.2690(45)$$<BR>\n" );
document.write( "$$\hat{y} = -0.9495 + 12.105$$<BR>\n" );
document.write( "$$\hat{y} \approx 11.16\%$$\r<BR>\n" );
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document.write( "**Estimated moisture content:** **11.16%**\r<BR>\n" );
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document.write( "### (iv) Correlation Coefficient ($r$)<BR>\n" );
document.write( "The correlation coefficient measures the strength and direction of the linear relationship.<BR>\n" );
document.write( "* **$r \approx 0.9144$**\r<BR>\n" );
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document.write( "Since $r$ is very close to $1$, it indicates a **very strong positive linear correlation** between humidity and moisture content. </SPAN>\n" );
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