document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168177>Question 1168177</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Waterbury Insurance Company wants to study the relationship between the amount of fire damage and the distance between the burning house and the nearest fire station. This information will be used in setting rates for insurance coverage. For a sample of 30 claims for the last year, the director of the actuarial department determined the distance from the fire station (x) and the amount of fire damage, in thousands of dollars (y). The MegaStat output is reported below.<BR>\n" );
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document.write( "ANOVA table<BR>\n" );
document.write( "Source	SS	df	MS	F<BR>\n" );
document.write( "Regression	1,830.5782	1	1,830.5782	 	41.15<BR>\n" );
document.write( "Residual	1,245.4934	28	44.4819	 	 <BR>\n" );
document.write( "Total	3,076.0716	29	 	 	 	 <BR>\n" );
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document.write( "Regression output<BR>\n" );
document.write( "Variables	Coefficients	Std. Error	t(df=28)<BR>\n" );
document.write( "Intercept	14.1988	3.137	2.962<BR>\n" );
document.write( "Distance  -  X	3.9798	8.842	6.41\r<BR>\n" );
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document.write( "a-1. Write out the regression equation. (Round your answers to 3 decimal places.)\r<BR>\n" );
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document.write( "b. How much damage would you estimate for a fire 9 miles from the nearest fire station? (Round your answer to the nearest dollar amount.)\r<BR>\n" );
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document.write( "c-1. Determine the coefficient of determination. (Round your answer to 3 decimal places.)\r<BR>\n" );
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document.write( "c-2. Fill in the blank below. (Round your answer to one decimal place.)\r<BR>\n" );
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document.write( "d-1. Determine the correlation coefficient. (Round your answer to 3 decimal places.)\r<BR>\n" );
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document.write( "d-3. How did you determine the sign of the correlation coefficient?\r<BR>\n" );
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document.write( "e-1. State the decision rule for 0.01 significance level: H0 : ρ = 0; H1 : ρ ≠ 0. (Negative value should be indicated by a minus sign. Round your answers to 3 decimal places.)\r<BR>\n" );
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document.write( "e-2. Compute the value of the test statistic for the hypothesis of β1. (Round your answer to 2 decimal places.)\r<BR>\n" );
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document.write( "e-3. Is there any significant relationship between the distance from the fire station and the amount of damage? Use the 0.01 significance level.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851670 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=851670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the MegaStat output to answer each part of the question.\r<BR>\n" );
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document.write( "**a-1. Write out the regression equation. (Round your answers to 3 decimal places.)**\r<BR>\n" );
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document.write( "The regression equation predicts the amount of fire damage (y) based on the distance from the fire station (x). From the \"Regression output\" section, we can identify the coefficients:\r<BR>\n" );
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document.write( "* Intercept (b₀) = 14.1988<BR>\n" );
document.write( "* Coefficient for Distance (b₁) = 3.9798\r<BR>\n" );
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document.write( "Rounding these coefficients to three decimal places, we get:\r<BR>\n" );
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document.write( "* Intercept (b₀) ≈ 14.199<BR>\n" );
document.write( "* Coefficient for Distance (b₁) ≈ 3.980\r<BR>\n" );
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document.write( "The regression equation is:<BR>\n" );
document.write( "$\hat{y} = b_0 + b_1 x$<BR>\n" );
document.write( "$\hat{y} = 14.199 + 3.980 x$\r<BR>\n" );
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document.write( "**b. How much damage would you estimate for a fire 9 miles from the nearest fire station? (Round your answer to the nearest dollar amount.)**\r<BR>\n" );
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document.write( "To estimate the damage for a fire 9 miles away, we substitute $x = 9$ into the regression equation:\r<BR>\n" );
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document.write( "$\hat{y} = 14.199 + 3.980 (9)$<BR>\n" );
document.write( "$\hat{y} = 14.199 + 35.820$<BR>\n" );
document.write( "$\hat{y} = 50.019$\r<BR>\n" );
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document.write( "Since $\hat{y}$ represents the amount of fire damage in thousands of dollars, we multiply by 1000 to get the damage in dollars:\r<BR>\n" );
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document.write( "Estimated damage = $50.019 \times 1000 = 50019$\r<BR>\n" );
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document.write( "Rounding to the nearest dollar amount, the estimated damage is $50,019.\r<BR>\n" );
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document.write( "**c-1. Determine the coefficient of determination. (Round your answer to 3 decimal places.)**\r<BR>\n" );
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document.write( "The coefficient of determination ($R^2$) measures the proportion of the variance in the dependent variable (fire damage) that is predictable from the independent variable (distance). It is calculated as:\r<BR>\n" );
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document.write( "$R^2 = \frac{SS_{Regression}}{SS_{Total}}$\r<BR>\n" );
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document.write( "From the ANOVA table:\r<BR>\n" );
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document.write( "* $SS_{Regression} = 1,830.5782$<BR>\n" );
document.write( "* $SS_{Total} = 3,076.0716$\r<BR>\n" );
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document.write( "$R^2 = \frac{1,830.5782}{3,076.0716} \approx 0.59509$\r<BR>\n" );
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document.write( "Rounding to three decimal places, the coefficient of determination is $0.595$.\r<BR>\n" );
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document.write( "**c-2. Fill in the blank below. (Round your answer to one decimal place.)**\r<BR>\n" );
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document.write( "The coefficient of determination indicates that approximately $\underline{59.5}$% of the variation in the amount of fire damage can be explained by the distance from the nearest fire station.\r<BR>\n" );
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document.write( "**d-1. Determine the correlation coefficient. (Round your answer to 3 decimal places.)**\r<BR>\n" );
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document.write( "The correlation coefficient ($r$) measures the strength and direction of the linear relationship between the two variables. It is the square root of the coefficient of determination, with the sign determined by the sign of the slope coefficient in the regression equation.\r<BR>\n" );
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document.write( "$r = \pm \sqrt{R^2}$<BR>\n" );
document.write( "$r = \pm \sqrt{0.595}$<BR>\n" );
document.write( "$r \approx \pm 0.77136$\r<BR>\n" );
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document.write( "Rounding to three decimal places, $r \approx \pm 0.771$.\r<BR>\n" );
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document.write( "**d-3. How did you determine the sign of the correlation coefficient?**\r<BR>\n" );
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document.write( "The sign of the correlation coefficient is the same as the sign of the coefficient for the independent variable (Distance - X) in the regression equation. In the \"Regression output\", the coefficient for Distance - X is 3.9798, which is positive. Therefore, the correlation coefficient is positive.\r<BR>\n" );
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document.write( "So, $r \approx +0.771$.\r<BR>\n" );
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document.write( "**e-1. State the decision rule for 0.01 significance level: H0 : ρ = 0; H1 : ρ ≠ 0. (Negative value should be indicated by a minus sign. Round your answers to 3 decimal places.)**\r<BR>\n" );
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document.write( "The decision rule for a two-tailed t-test for the correlation coefficient at a significance level of $\alpha = 0.01$ and degrees of freedom $df = n - 2 = 30 - 2 = 28$ is to reject the null hypothesis ($H_0: \rho = 0$) if the absolute value of the test statistic ($|t|$) is greater than the critical t-value ($t_{\alpha/2, df}$).\r<BR>\n" );
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document.write( "Looking up the critical t-value for $\alpha/2 = 0.01 / 2 = 0.005$ and $df = 28$ in a t-distribution table, we find $t_{0.005, 28} \approx 2.763$.\r<BR>\n" );
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document.write( "**Decision Rule:** Reject $H_0$ if $|t| > 2.763$.\r<BR>\n" );
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document.write( "**e-2. Compute the value of the test statistic for the hypothesis of β1. (Round your answer to 2 decimal places.)**\r<BR>\n" );
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document.write( "The test statistic for the hypothesis of $\beta_1$ (the slope coefficient) is given by the t-value in the \"Regression output\" for the \"Distance - X\" variable.\r<BR>\n" );
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document.write( "$t = 6.41$\r<BR>\n" );
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document.write( "Rounding to two decimal places, the test statistic is $6.41$.\r<BR>\n" );
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document.write( "**e-3. Is there any significant relationship between the distance from the fire station and the amount of damage? Use the 0.01 significance level.**\r<BR>\n" );
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document.write( "To determine if there is a significant relationship, we compare the absolute value of the test statistic to the critical t-value from the decision rule in e-1.\r<BR>\n" );
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document.write( "$|t| = |6.41| = 6.41$<BR>\n" );
document.write( "Critical t-value = $2.763$\r<BR>\n" );
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document.write( "Since $6.41 > 2.763$, we reject the null hypothesis ($H_0: \rho = 0$ or equivalently $H_0: \beta_1 = 0$).\r<BR>\n" );
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document.write( "**Conclusion:** There is a statistically significant relationship between the distance from the fire station and the amount of fire damage at the 0.01 significance level.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{a-1. \hat{y} = 14.199 + 3.980 x, b. \$50019, c-1. 0.595, c-2. 59.5, d-1. 0.771, d-3. The sign is the same as the positive slope coefficient, e-1. Reject H0 if |t| > 2.763, e-2. 6.41, e-3. Yes}$ </SPAN>\n" );
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