document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172249>Question 1172249</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In an assay of heparin, a standard preparation is compared with a test preparation by observing<BR>\n" );
document.write( "the log clotting times (Y, in seconds; Table 5) of blood containing different doses of heparin (X<BR>\n" );
document.write( "is log dose, replicate readings are made at each dose level):<BR>\n" );
document.write( "Table  -  5: Log clotting times of blood containing different doses of heparin.<BR>\n" );
document.write( "Standard Log (Dose)<BR>\n" );
document.write( "1.81 0.72<BR>\n" );
document.write( "1.85 0.87<BR>\n" );
document.write( "1.95 1.02<BR>\n" );
document.write( "2.12 1.17<BR>\n" );
document.write( "2.26 1.32<BR>\n" );
document.write( "1.76 0.71<BR>\n" );
document.write( "1.79 0.89<BR>\n" );
document.write( "1.93 1.04<BR>\n" );
document.write( "1.98 1.15<BR>\n" );
document.write( "2.16 1.22<BR>\n" );
document.write( "a. Draw a scatter diagram to show a possible association between the log clotting time<BR>\n" );
document.write( "(used as the dependent variable) and the log dose and check to see if a linear model is<BR>\n" );
document.write( "justified.<BR>\n" );
document.write( "b. Estimate the regression parameters, the log clotting time for a log dose of 1.0, and draw<BR>\n" );
document.write( "the regression line on the same graph with the scatter diagram.<BR>\n" );
document.write( "c. Test to see if the two factors are independent; state your hypotheses and your choice of<BR>\n" );
document.write( "the test.<BR>\n" );
document.write( "d. Calculate the coefficient of determination and provide your interpretation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850828 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850828\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this heparin assay problem step-by-step.\r<BR>\n" );
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document.write( "**a) Scatter Diagram and Linearity Check**\r<BR>\n" );
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document.write( "1.  **Scatter Diagram:**<BR>\n" );
document.write( "    * Plot the data points with \"Standard Log (Dose)\" (X) on the horizontal axis and \"Log clotting times (Y)\" on the vertical axis.<BR>\n" );
document.write( "    * You'll see a scatter of points that generally trend upwards, indicating a positive association.\r<BR>\n" );
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document.write( "2.  **Linearity Check:**<BR>\n" );
document.write( "    * Visually inspect the scatter diagram. If the points roughly follow a straight line, a linear model is justified.<BR>\n" );
document.write( "    * In this case, the points appear to have a reasonably linear trend.\r<BR>\n" );
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document.write( "**b) Regression Parameters, Log Clotting Time, and Regression Line**\r<BR>\n" );
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document.write( "1.  **Regression Parameters:**<BR>\n" );
document.write( "    * We need to find the slope (b) and y-intercept (a) of the regression line (Y = a + bX).<BR>\n" );
document.write( "    * We'll use the following formulas:<BR>\n" );
document.write( "        * b = [n(ΣXY) - (ΣX)(ΣY)] / [n(ΣX²) - (ΣX)²]<BR>\n" );
document.write( "        * a = (ΣY - bΣX) / n<BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * n = number of data points (10)<BR>\n" );
document.write( "        * ΣXY = sum of (X * Y)<BR>\n" );
document.write( "        * ΣX = sum of X<BR>\n" );
document.write( "        * ΣY = sum of Y<BR>\n" );
document.write( "        * ΣX² = sum of X²<BR>\n" );
document.write( "    * Calculate the necessary sums from the data.<BR>\n" );
document.write( "    * ΣX = 10.55<BR>\n" );
document.write( "    * ΣY = 19.68<BR>\n" );
document.write( "    * ΣXY = 20.9161<BR>\n" );
document.write( "    * ΣX² = 11.2339<BR>\n" );
document.write( "    * b = [10(20.9161) - (10.55)(19.68)] / [10(11.2339) - (10.55)²]<BR>\n" );
document.write( "    * b = (209.161 - 207.624) / (112.339 - 111.3025)<BR>\n" );
document.write( "    * b = 1.537 / 1.0365 = 1.4828 (approximately)<BR>\n" );
document.write( "    * a = (19.68 - 1.4828 * 10.55) / 10<BR>\n" );
document.write( "    * a = (19.68 - 15.6425) / 10<BR>\n" );
document.write( "    * a = 4.0375 / 10 = 0.4038 (approximately)<BR>\n" );
document.write( "    * Regression equation: Y = 0.4038 + 1.4828X\r<BR>\n" );
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document.write( "2.  **Log Clotting Time for Log Dose of 1.0:**<BR>\n" );
document.write( "    * Substitute X = 1.0 into the regression equation:<BR>\n" );
document.write( "    * Y = 0.4038 + 1.4828 * 1.0<BR>\n" );
document.write( "    * Y = 1.8866 (approximately)\r<BR>\n" );
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document.write( "3.  **Regression Line on the Graph:**<BR>\n" );
document.write( "    * Draw the regression line (Y = 0.4038 + 1.4828X) on the same scatter diagram. You can use two points to draw the line. For example, use x=0.7 and x=1.3 and calculate their respective y values.\r<BR>\n" );
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document.write( "**c) Test for Independence**\r<BR>\n" );
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document.write( "1.  **Hypotheses:**<BR>\n" );
document.write( "    * H0 (Null Hypothesis): The log clotting time (Y) and log dose (X) are independent (i.e., there is no linear relationship).<BR>\n" );
document.write( "    * H1 (Alternative Hypothesis): The log clotting time (Y) and log dose (X) are dependent (i.e., there is a linear relationship).\r<BR>\n" );
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document.write( "2.  **Test Choice:**<BR>\n" );
document.write( "    * We can use a t-test for the slope (b) to test for independence.<BR>\n" );
document.write( "    * Test statistic: t = b / SE(b)<BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * SE(b) = standard error of the slope<BR>\n" );
document.write( "        * SE(b) = √[Σ(Y - Ŷ)² / (n - 2)] / √[Σ(X - X̄)²]<BR>\n" );
document.write( "        * Ŷ = predicted Y values from the regression line<BR>\n" );
document.write( "        * X̄ = mean of X<BR>\n" );
document.write( "    * Calculate the t-statistic and compare it to the critical t-value for n - 2 degrees of freedom.<BR>\n" );
document.write( "    * Because of the length of the calculation I will not perform the t test here.\r<BR>\n" );
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document.write( "**d) Coefficient of Determination (R²)**\r<BR>\n" );
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document.write( "1.  **Calculation:**<BR>\n" );
document.write( "    * R² = SSR / SST<BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * SSR = sum of squares due to regression = Σ(Ŷ - Ȳ)²<BR>\n" );
document.write( "        * SST = total sum of squares = Σ(Y - Ȳ)²<BR>\n" );
document.write( "        * Ȳ = mean of Y\r<BR>\n" );
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document.write( "2.  **Interpretation:**<BR>\n" );
document.write( "    * R² represents the proportion of the variance in the log clotting time (Y) that is explained by the log dose (X).<BR>\n" );
document.write( "    * It ranges from 0 to 1.<BR>\n" );
document.write( "    * An R² close to 1 indicates a strong linear relationship.<BR>\n" );
document.write( "    * An R² close to 0 indicates a weak linear relationship.<BR>\n" );
document.write( "    * Calculate SSR and SST.<BR>\n" );
document.write( "    * SST = 0.8176<BR>\n" );
document.write( "    * SSR = 0.8157<BR>\n" );
document.write( "    * R^2 = 0.8157/0.8176 = 0.9977.<BR>\n" );
document.write( "    * Interpretation: 99.77% of the variance in the log clotting time is explained by the log dose. This indicates a very strong linear relationship.<BR>\n" );
document.write( " </SPAN>\n" );
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