document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178849>Question 1178849</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You are given the following sample data for variables x and y:<BR>\n" );
document.write( "X (independent) |    Y (dependent)|<BR>\n" );
document.write( "1               |     16          |<BR>\n" );
document.write( "7               |     50          |<BR>\n" );
document.write( "3               |     22          |<BR>\n" );
document.write( "8               |     59          |<BR>\n" );
document.write( "11              |     63          |<BR>\n" );
document.write( "5               |     46          |<BR>\n" );
document.write( "4               |     43          |\r<BR>\n" );
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document.write( "1. Construct a scatter plot for these data and describe what, if any, relationship<BR>\n" );
document.write( "appears to exist.<BR>\n" );
document.write( "2. Calculate product-moment correlation and verify the answer by relating to<BR>\n" );
document.write( "scatter plot you derived from the previous question.<BR>\n" );
document.write( "3. Compute the regression equation based on these sample data and interpret the<BR>\n" );
document.write( "regression coefficients.<BR>\n" );
document.write( "4. Based on the sample data, what percentage of the total variation in the dependent<BR>\n" );
document.write( "the variable can be explained by the independent variable?<BR>\n" );
document.write( "5.Explain why spearman rank correlation coefficient is not suitable to analyze<BR>\n" );
document.write( "above data set? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850287 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=850287\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step:\r<BR>\n" );
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document.write( "**1. Construct a Scatter Plot and Describe the Relationship:**\r<BR>\n" );
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document.write( "* **Scatter Plot:**<BR>\n" );
document.write( "    * Plot the points (1, 16), (7, 50), (3, 22), (8, 59), (11, 63), (5, 46), and (4, 43) on a graph with x on the horizontal axis and y on the vertical axis.<BR>\n" );
document.write( "    * You'll observe that as x increases, y tends to increase as well.<BR>\n" );
document.write( "* **Relationship:**<BR>\n" );
document.write( "    * There appears to be a **positive linear relationship** between x and y. As the independent variable (x) increases, the dependent variable (y) also tends to increase. The points roughly follow a straight line pattern.\r<BR>\n" );
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document.write( "**2. Calculate Product-Moment Correlation (Pearson's r):**\r<BR>\n" );
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document.write( "* **Formula:**<BR>\n" );
document.write( "    * r = [n(∑xy) - (∑x)(∑y)] / √{[n∑x² - (∑x)²][n∑y² - (∑y)²]}<BR>\n" );
document.write( "* **Calculations:**<BR>\n" );
document.write( "    * n = 7<BR>\n" );
document.write( "    * ∑x = 1 + 7 + 3 + 8 + 11 + 5 + 4 = 39<BR>\n" );
document.write( "    * ∑y = 16 + 50 + 22 + 59 + 63 + 46 + 43 = 299<BR>\n" );
document.write( "    * ∑x² = 1² + 7² + 3² + 8² + 11² + 5² + 4² = 275<BR>\n" );
document.write( "    * ∑y² = 16² + 50² + 22² + 59² + 63² + 46² + 43² = 14,795<BR>\n" );
document.write( "    * ∑xy = (1 * 16) + (7 * 50) + (3 * 22) + (8 * 59) + (11 * 63) + (5 * 46) + (4 * 43) = 1,939<BR>\n" );
document.write( "    * r = [7(1939) - (39)(299)] / √{[7(275) - (39)²][7(14795) - (299)²]}<BR>\n" );
document.write( "    * r = [13573 - 11661] / √{[1925 - 1521][103565 - 89401]}<BR>\n" );
document.write( "    * r = 1912 / √[404 * 14164]<BR>\n" );
document.write( "    * r = 1912 / √5723256<BR>\n" );
document.write( "    * r = 1912 / 2392.33<BR>\n" );
document.write( "    * r ≈ 0.7992<BR>\n" );
document.write( "* **Verification:**<BR>\n" );
document.write( "    * The calculated r (0.7992) is positive and relatively close to 1, indicating a strong positive linear relationship. This aligns with the scatter plot, which shows a positive linear trend.\r<BR>\n" );
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document.write( "**3. Compute the Regression Equation and Interpret Coefficients:**\r<BR>\n" );
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document.write( "* **Regression Equation:** y = a + bx<BR>\n" );
document.write( "* **Calculate b (slope):**<BR>\n" );
document.write( "    * b = [n(∑xy) - (∑x)(∑y)] / [n(∑x²) - (∑x)²]<BR>\n" );
document.write( "    * b = 1912 / 404<BR>\n" );
document.write( "    * b ≈ 4.7327<BR>\n" );
document.write( "* **Calculate a (y-intercept):**<BR>\n" );
document.write( "    * a = (∑y / n) - b(∑x / n)<BR>\n" );
document.write( "    * a = (299 / 7) - 4.7327(39 / 7)<BR>\n" );
document.write( "    * a ≈ 42.7143 - 26.3379<BR>\n" );
document.write( "    * a ≈ 16.3764<BR>\n" );
document.write( "* **Regression Equation:** y = 16.3764 + 4.7327x<BR>\n" );
document.write( "* **Interpretation:**<BR>\n" );
document.write( "    * **b (4.7327):** For every one-unit increase in x, y is predicted to increase by approximately 4.7327 units.<BR>\n" );
document.write( "    * **a (16.3764):** When x is 0, the predicted value of y is approximately 16.3764.\r<BR>\n" );
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document.write( "**4. Percentage of Total Variation Explained (R²):**\r<BR>\n" );
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document.write( "* **R² = r²** (coefficient of determination)<BR>\n" );
document.write( "* R² = (0.7992)²<BR>\n" );
document.write( "* R² ≈ 0.6387<BR>\n" );
document.write( "* **Percentage:** 0.6387 * 100% ≈ 63.87%<BR>\n" );
document.write( "* **Interpretation:** Approximately 63.87% of the total variation in y can be explained by the linear relationship with x.\r<BR>\n" );
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document.write( "**5. Why Spearman Rank Correlation is Not Suitable:**\r<BR>\n" );
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document.write( "* **Spearman Rank Correlation:** This coefficient measures the strength and direction of a monotonic relationship (not necessarily linear) between two ranked variables. It's used when data might not be normally distributed or when relationships are not strictly linear.<BR>\n" );
document.write( "* **Suitability for Pearson's r:** The data appears to have a relatively strong linear relationship, as observed in the scatter plot and confirmed by a high Pearson's r value (0.7992).<BR>\n" );
document.write( "* **Unnecessary Ranking:** Spearman's rank correlation requires ranking the data. If the relationship is adequately linear, ranking the data loses the magnitude of the differences between the values.<BR>\n" );
document.write( "* **Pearson's r is More Powerful:** For linear relationships and normally distributed data (or at least data that don't violate the assumptions of linear regression), Pearson's r is generally considered more powerful and informative than Spearman's rank correlation.<BR>\n" );
document.write( "* **Conclusion:** Since the scatter plot suggests a linear relationship, and Pearson's r shows a strong linear correlation, there is no need to rank the data. Pearson's r is more appropriate in this scenario.<BR>\n" );
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