document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163537>Question 1163537</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A realtor studies the relationship between the size of a house (in square feet) and the property taxes (in $) owed by the owner. The table below shows a portion of the data for 20 homes in a suburb 60 miles outside of New York City. [You may find it useful to reference the t table.] \r<BR>\n" );
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document.write( "Property <BR>\n" );
document.write( "Taxes	Size<BR>\n" );
document.write( "21860	2486<BR>\n" );
document.write( "17463	2447<BR>\n" );
document.write( "18232	1851<BR>\n" );
document.write( "15691	1000<BR>\n" );
document.write( "43998	5688<BR>\n" );
document.write( "33674	2547<BR>\n" );
document.write( "15233	2221<BR>\n" );
document.write( "16760	1913<BR>\n" );
document.write( "18247	2013<BR>\n" );
document.write( "16777	1368<BR>\n" );
document.write( "15164	1393<BR>\n" );
document.write( "36087	3099<BR>\n" );
document.write( "31021	2808<BR>\n" );
document.write( "42137	3322<BR>\n" );
document.write( "14374	1582<BR>\n" );
document.write( "38935	4096<BR>\n" );
document.write( "25336	4047<BR>\n" );
document.write( "22947	2471<BR>\n" );
document.write( "16177	3551<BR>\n" );
document.write( "29226	2845\r<BR>\n" );
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document.write( "a-1. Calculate the sample correlation coefficient rxy. (Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.) \r<BR>\n" );
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document.write( "a-2. Interpret rxy. \r<BR>\n" );
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document.write( " a. The correlation coefficient indicates a positive linear relationship.<BR>\n" );
document.write( " b. The correlation coefficient indicates a negative linear relationship.<BR>\n" );
document.write( " c. The correlation coefficient indicates no linear relationship. \r<BR>\n" );
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document.write( "b. Specify the competing hypotheses in order to determine whether the population correlation coefficient between the size of a house and property taxes differs from zero. \r<BR>\n" );
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document.write( " a. H0: ρxy = 0; HA: ρxy ≠ 0<BR>\n" );
document.write( " b. H0: ρxy ≥ 0; HA: ρxy < 0<BR>\n" );
document.write( " c. H0: ρxy ≤ 0; HA: ρxy > 0\r<BR>\n" );
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document.write( " <BR>\n" );
document.write( "c-1. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)\r<BR>\n" );
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document.write( " \r<BR>\n" );
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document.write( "c-2. Find the p-value.\r<BR>\n" );
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document.write( " a. p-value < 0.01<BR>\n" );
document.write( " b. p-value > or equal to 0.10<BR>\n" );
document.write( " c. 0.05 < or equal to p-value < 0.10<BR>\n" );
document.write( " d. 0.02 < or equal to p-value < 0.05<BR>\n" );
document.write( " e. 0.01 < or equal to p-value < 0.02\r<BR>\n" );
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document.write( "d. At the 1% significance level, what is the conclusion to the test?\r<BR>\n" );
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document.write( " a.  Reject H0; we can state size and property taxes are correlated.<BR>\n" );
document.write( " b.  Reject H0; we cannot state size and property taxes are correlated.<BR>\n" );
document.write( " c.  Do not reject H0; we can state size and property taxes are correlated.<BR>\n" );
document.write( " d.  Do not reject H0; we cannot state size and property taxes are correlated.<BR>\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 #854375 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.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=854375\">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( "import scipy.stats as stats\r<BR>\n" );
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document.write( "# Data<BR>\n" );
document.write( "taxes = np.array([21860, 17463, 18232, 15691, 43998, 33674, 15233, 16760, 18247, 16777, 15164, 36087, 31021, 42137, 14374, 38935, 25336, 22947, 16177, 29226])<BR>\n" );
document.write( "size = np.array([2486, 2447, 1851, 1000, 5688, 2547, 2221, 1913, 2013, 1368, 1393, 3099, 2808, 3322, 1582, 4096, 4047, 2471, 3551, 2845])\r<BR>\n" );
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document.write( "# n<BR>\n" );
document.write( "n = len(taxes)\r<BR>\n" );
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document.write( "# r_xy<BR>\n" );
document.write( "r, p_val = stats.pearsonr(size, taxes)\r<BR>\n" );
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document.write( "# t-statistic for correlation<BR>\n" );
document.write( "# t = r * sqrt(n-2) / sqrt(1-r^2)<BR>\n" );
document.write( "t_stat = r * np.sqrt(n-2) / np.sqrt(1 - r**2)\r<BR>\n" );
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document.write( "print(f\"Sample size n: {n}\")<BR>\n" );
document.write( "print(f\"Correlation coefficient r_xy: {r:.4f}\")<BR>\n" );
document.write( "print(f\"Test statistic t: {t_stat:.3f}\")<BR>\n" );
document.write( "print(f\"p-value: {p_val:.10f}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Sample size n: 20<BR>\n" );
document.write( "Correlation coefficient r_xy: 0.7587<BR>\n" );
document.write( "Test statistic t: 4.940<BR>\n" );
document.write( "p-value: 0.0001056345\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "Based on the data provided for the 20 homes, here is the statistical analysis of the relationship between house size and property taxes:\r<BR>\n" );
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document.write( "### a-1. Sample Correlation Coefficient $r_{xy}$<BR>\n" );
document.write( "Using the formula for the Pearson correlation coefficient:<BR>\n" );
document.write( "$$r_{xy} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$$<BR>\n" );
document.write( "The calculated value is:<BR>\n" );
document.write( "**$r_{xy} = 0.7587$**\r<BR>\n" );
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document.write( "### a-2. Interpretation of $r_{xy}$<BR>\n" );
document.write( "Since $0.7587$ is a positive value, it indicates that as the size of the house increases, the property taxes also tend to increase.<BR>\n" );
document.write( "**Correct Choice: a. The correlation coefficient indicates a positive linear relationship.**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### b. Competing Hypotheses<BR>\n" );
document.write( "To determine if the population correlation coefficient ($\rho_{xy}$) differs from zero (indicating a significant relationship), we set up a two-tailed test:<BR>\n" );
document.write( "* **$H_0: \rho_{xy} = 0$** (No correlation)<BR>\n" );
document.write( "* **$H_A: \rho_{xy} \neq 0$** (Correlation exists)\r<BR>\n" );
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document.write( "**Correct Choice: a.**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### c-1. Test Statistic<BR>\n" );
document.write( "The test statistic for the correlation coefficient follows a $t$-distribution with $n - 2$ degrees of freedom ($df = 18$):<BR>\n" );
document.write( "$$t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} = \frac{0.7587\sqrt{18}}{\sqrt{1-0.7587^2}} \approx 4.940$$<BR>\n" );
document.write( "**Value of the test statistic: $4.940$**\r<BR>\n" );
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document.write( "### c-2. $p$-value<BR>\n" );
document.write( "For a $t$-statistic of $4.940$ with $18$ degrees of freedom, the $p$-value is approximately $0.0001$.<BR>\n" );
document.write( "**Correct Choice: a. $p\text{-value} < 0.01$**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### d. Conclusion to the Test<BR>\n" );
document.write( "At the **1% significance level** ($\alpha = 0.01$), we compare the $p$-value to $\alpha$:<BR>\n" );
document.write( "* Since $0.0001 < 0.01$, we **reject the null hypothesis ($H_0$)**.<BR>\n" );
document.write( "* There is sufficient evidence to conclude that a linear relationship exists between house size and property taxes.\r<BR>\n" );
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document.write( "**Correct Choice: a. Reject $H_0$; we can state size and property taxes are correlated.** </SPAN>\n" );
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