document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180053>Question 1180053</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Graph the following residuals, and indicate which of the assumptions underlying regression appear to be in jeopardy on the basis of the graph:\r<BR>\n" );
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document.write( "x=    213  216  227  229  237  247  263 <BR>\n" );
document.write( "y- ŷ=  -  11  -  5  -  2  -  1    6   10   12 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850127 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=850127\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to graph the residuals and analyze the assumptions:\r<BR>\n" );
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document.write( "1. **Create a Scatterplot:**\r<BR>\n" );
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document.write( "* Plot the x-values on the horizontal axis and the residuals (y - ŷ) on the vertical axis.\r<BR>\n" );
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document.write( "2. **Analyze the Scatterplot:**\r<BR>\n" );
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document.write( "* **Linearity:** Look for a random scatter of points around the horizontal axis (y = 0). If there is a clear pattern (e.g., a curve), the linearity assumption might be violated.<BR>\n" );
document.write( "* **Constant Variance (Homoscedasticity):** The spread of the residuals should be roughly constant across all x-values. If the spread increases or decreases as x increases, the constant variance assumption might be violated.<BR>\n" );
document.write( "* **Normality:** While harder to assess visually, look for a roughly symmetric distribution of residuals around the horizontal axis. If the distribution is heavily skewed or has outliers, the normality assumption might be in jeopardy.<BR>\n" );
document.write( "* **Independence:** This assumption is difficult to assess from a residual plot alone. However, if there is a clear pattern (e.g., a cyclical trend), it could suggest that the residuals are not independent.\r<BR>\n" );
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document.write( "**Analysis of Your Data:**\r<BR>\n" );
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document.write( "Based on the given data, here's what we can observe from a scatterplot of the residuals:\r<BR>\n" );
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document.write( "* **Linearity:** There seems to be a slight upward trend in the residuals as x increases. This could indicate a mild violation of the linearity assumption.<BR>\n" );
document.write( "* **Constant Variance:** The spread of residuals appears to be relatively constant across the x-values, suggesting that the constant variance assumption is likely met.<BR>\n" );
document.write( "* **Normality:** With only seven data points, it's difficult to assess normality visually. However, there are no extreme outliers or strong skewness.<BR>\n" );
document.write( "* **Independence:** There is no clear pattern suggesting a violation of independence.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "Based on the residual plot, the linearity assumption might be slightly in jeopardy. However, with such a small sample size, it's difficult to draw definitive conclusions. Further analysis with more data points might be needed to confirm any violations. </SPAN>\n" );
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