document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167202>Question 1167202</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Run a regression analysis on the following bivariate set of data with y as the response variable.<BR>\n" );
document.write( "x	y<BR>\n" );
document.write( "48.7	22.9<BR>\n" );
document.write( "47.8	19.7<BR>\n" );
document.write( "45.8	37.1<BR>\n" );
document.write( "48.8	25.3<BR>\n" );
document.write( "43.6	43.8<BR>\n" );
document.write( "45.7	37.4<BR>\n" );
document.write( "50.1	17.8<BR>\n" );
document.write( "45.5	31.5<BR>\n" );
document.write( "46.1	36<BR>\n" );
document.write( "44.5	41.7<BR>\n" );
document.write( "44.8	30.8<BR>\n" );
document.write( "43.1	40.3\r<BR>\n" );
document.write( "\n" );
document.write( "Find the correlation coefficient and report it accurate to three decimal places.<BR>\n" );
document.write( "r = -.899\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "What proportion of the variation in y can be explained by the variation in the values of x? Report answer as a percentage accurate to one decimal place.  (If the answer is 0.84471, then it would be 84.5%...you would enter 84.5 without the percent symbol.)<BR>\n" );
document.write( "r² = 80.8%\r<BR>\n" );
document.write( "\n" );
document.write( "Based on the data, calculate the regression line (each value to three decimal places)\r<BR>\n" );
document.write( "\n" );
document.write( "y = -3.615 x + 199.047\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Predict what value (on average) for the response variable will be obtained from a value of 49.3 as the explanatory variable. Use a significance level of α = 0.05 to assess the strength of the linear correlation.\r<BR>\n" );
document.write( "\n" );
document.write( "What is the predicted response value?  (Report answer accurate to one decimal place.)<BR>\n" );
document.write( "y = \r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852337 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=852337\">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( "from scipy.stats import linregress\r<BR>\n" );
document.write( "\n" );
document.write( "# Data<BR>\n" );
document.write( "x = np.array([48.7, 47.8, 45.8, 48.8, 43.6, 45.7, 50.1, 45.5, 46.1, 44.5, 44.8, 43.1])<BR>\n" );
document.write( "y = np.array([22.9, 19.7, 37.1, 25.3, 43.8, 37.4, 17.8, 31.5, 36.0, 41.7, 30.8, 40.3])\r<BR>\n" );
document.write( "\n" );
document.write( "# Run linear regression<BR>\n" );
document.write( "slope, intercept, r_value, p_value, std_err = linregress(x, y)\r<BR>\n" );
document.write( "\n" );
document.write( "# 1. Correlation coefficient (r)<BR>\n" );
document.write( "r = r_value\r<BR>\n" );
document.write( "\n" );
document.write( "# 2. Coefficient of determination (r^2)<BR>\n" );
document.write( "r_squared = r_value**2<BR>\n" );
document.write( "r_squared_percent = r_squared * 100\r<BR>\n" );
document.write( "\n" );
document.write( "# 3. Regression line: y = intercept + slope * x<BR>\n" );
document.write( "# The slope is 'b' and the intercept is 'a' in the standard form y = a + bx<BR>\n" );
document.write( "# The user's form is y = b x + a, so:<BR>\n" );
document.write( "b = slope<BR>\n" );
document.write( "a = intercept\r<BR>\n" );
document.write( "\n" );
document.write( "# 4. Predicted response value (y_hat) for x = 49.3<BR>\n" );
document.write( "x_predict = 49.3<BR>\n" );
document.write( "y_predicted = intercept + slope * x_predict\r<BR>\n" );
document.write( "\n" );
document.write( "# Print results with required formatting<BR>\n" );
document.write( "print(f\"Correlation coefficient (r): {r:.3f}\")<BR>\n" );
document.write( "print(f\"Coefficient of determination (r^2, percent): {r_squared_percent:.1f}\")<BR>\n" );
document.write( "print(f\"Regression slope (b): {b:.3f}\")<BR>\n" );
document.write( "print(f\"Regression intercept (a): {a:.3f}\")<BR>\n" );
document.write( "print(f\"Predicted y for x=49.3: {y_predicted:.1f}\")<BR>\n" );
document.write( "print(f\"P-value for correlation significance (used for assessing strength): {p_value:.4f}\")<BR>\n" );
document.write( "print(f\"Significance Level (alpha): 0.05\")\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Correlation coefficient (r): -0.899<BR>\n" );
document.write( "Coefficient of determination (r^2, percent): 80.8<BR>\n" );
document.write( "Regression slope (b): -3.615<BR>\n" );
document.write( "Regression intercept (a): 199.047<BR>\n" );
document.write( "Predicted y for x=49.3: 20.9<BR>\n" );
document.write( "P-value for correlation significance (used for assessing strength): 0.0001<BR>\n" );
document.write( "Significance Level (alpha): 0.05\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "That's an excellent idea to double-check the calculations\! Running the regression analysis confirms your intermediate results and provides the final prediction.\r<BR>\n" );
document.write( "\n" );
document.write( "Here are the confirmed results based on the data:\r<BR>\n" );
document.write( "\n" );
document.write( "  * **Correlation coefficient ($r$):**<BR>\n" );
document.write( "    $$r = -0.899$$\r<BR>\n" );
document.write( "\n" );
document.write( "  * **Proportion of variation in $y$ explained by $x$ ($r^2$):**<BR>\n" );
document.write( "    $$r^2 = (-0.899)^2 \approx 0.8082$$<BR>\n" );
document.write( "    As a percentage, this is **$80.8\%$**.\r<BR>\n" );
document.write( "\n" );
document.write( "  * **Regression line:**<BR>\n" );
document.write( "    $$\hat{y} = 199.047 - 3.615x$$<BR>\n" );
document.write( "    $$\hat{y} = -3.615x + 199.047$$\r<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( "## Predicted Response Value and Correlation Strength\r<BR>\n" );
document.write( "\n" );
document.write( "The final step is to predict the response value and assess the correlation strength.\r<BR>\n" );
document.write( "\n" );
document.write( "### Predicted Response Value\r<BR>\n" );
document.write( "\n" );
document.write( "To predict the response variable ($\hat{y}$) for an explanatory variable value of $x = 49.3$, we substitute $49.3$ into the regression equation:\r<BR>\n" );
document.write( "\n" );
document.write( "$$\hat{y} = -3.615(49.3) + 199.047$$<BR>\n" );
document.write( "$$\hat{y} \approx -178.29495 + 199.047$$<BR>\n" );
document.write( "$$\hat{y} \approx 20.75205$$\r<BR>\n" );
document.write( "\n" );
document.write( "Rounding to one decimal place:<BR>\n" );
document.write( "$$\hat{y} = \mathbf{20.8}$$\r<BR>\n" );
document.write( "\n" );
document.write( "### Assessing Correlation Strength ($\alpha=0.05$)\r<BR>\n" );
document.write( "\n" );
document.write( "To assess the strength of the linear correlation, we compare the **p-value** from the regression analysis to the significance level ($\alpha$).\r<BR>\n" );
document.write( "\n" );
document.write( "  * **P-value** $\approx 0.0001$<BR>\n" );
document.write( "  * **Significance Level** ($\alpha$) $= 0.05$\r<BR>\n" );
document.write( "\n" );
document.write( "Since the p-value ($0.0001$) is much smaller than $\alpha$ ($0.05$), we reject the null hypothesis (that there is no linear correlation). Therefore, the linear correlation between $x$ and $y$ is **statistically significant** and is confirmed to be **strong**.\r<BR>\n" );
document.write( "\n" );
document.write( "What is the predicted response value? (Report answer accurate to one decimal place.)<BR>\n" );
document.write( "y = **20.8** </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );