document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210185>Question 1210185</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The data for the two variables X and Y are given in the table below:<BR>\n" );
document.write( "X: 1.11, 0.00, 0.47, 0.23, 0.14, 0.29, 0.53, 0.61, 0.83, 0.65, 1.05, 0.31<BR>\n" );
document.write( "Y: 2.38, 1.03, 1.00, 0.90, 0.93, 0.90, 1.06, 1.16, 1.57, 1.22, 2.18, 0.91<BR>\n" );
document.write( "X: 1.35, 0.04, 1.03, 0.64, 0.86, 0.22, 0.30, 1.23, 1.49, 0.48, 1.07, 1.35<BR>\n" );
document.write( "Y: 3.32, 0.99, 2.12, 1.21, 1.65, 0.90, 0.91, 2.82, 3.98, 1.01, 2.25, 3.32<BR>\n" );
document.write( "Part I<BR>\n" );
document.write( "Given that X ~ U(0, θ) (i.e., X is uniformly distributed on [0, θ]), we can use the following estimators for θ:<BR>\n" );
document.write( "T₁ = 2X̄, where X̄ is the sample mean.<BR>\n" );
document.write( "T₂ = 2m̂, where m̂ is the sample median.<BR>\n" );
document.write( "T₃ = 2√3S, where S is the sample standard deviation.<BR>\n" );
document.write( "T₄ = max {X₁, X₂, …, Xₙ}, where the maximum value is taken from the sample.<BR>\n" );
document.write( "a) Compute the estimates T₁, T₂, T₃, T₄ for θ using the given sample data.<BR>\n" );
document.write( "b) Compare the standard errors of the above estimators.<BR>\n" );
document.write( "c) Assume that we additionally know that θ = 1 + e, with e ~ Exponential(1) and θ ≤ 2. Use the Bayesian inference technique to find θ and compare the error of this estimator with the ones above.<BR>\n" );
document.write( "Part II<BR>\n" );
document.write( "d) Calculate the sample correlation coefficient between X and Y.<BR>\n" );
document.write( "e) Test the hypothesis \"X and Y are correlated\".<BR>\n" );
document.write( "Part III<BR>\n" );
document.write( "We consider the linear regression model:<BR>\n" );
document.write( "  Y = a + bX + cX² + ε,<BR>\n" );
document.write( "where a, b, c are the coefficients and ε is the error term.<BR>\n" );
document.write( "f) Find the regression coefficients a, b, c.<BR>\n" );
document.write( "g) Construct 95% confidence intervals for a, b, and c.<BR>\n" );
document.write( "h) Assume that we use the above regression model to predict Y = y₀ at x₀ = 0.5. Calculate the prediction error and construct the 95% confidence interval for y₀.<BR>\n" );
document.write( "The answer provided by @CPhill is incomplete and incorrect. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851503 by </B><A HREF=/tutors/aboutme.mpl?userid=asinus><B>asinus(45)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=asinus><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851503\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's break down this problem step by step.\r<BR>\n" );
document.write( "\n" );
document.write( "**Part I: Estimating θ for X ~ U(0, θ)**\r<BR>\n" );
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document.write( "First, let's gather the X values:\r<BR>\n" );
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document.write( "X: 1.11, 0.00, 0.47, 0.23, 0.14, 0.29, 0.53, 0.61, 0.83, 0.65, 1.05, 0.31, 1.35, 0.04, 1.03, 0.64, 0.86, 0.22, 0.30, 1.23, 1.49, 0.48, 1.07, 1.35\r<BR>\n" );
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document.write( "There are n = 24 data points.\r<BR>\n" );
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document.write( "**a) Compute the estimates T₁, T₂, T₃, T₄ for θ.**\r<BR>\n" );
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document.write( "1.  **T₁ = 2X̄ (Twice the sample mean)**\r<BR>\n" );
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document.write( "    * Calculate the sum of X:<BR>\n" );
document.write( "        $$ \sum X = 1.11 + 0.00 + 0.47 + 0.23 + 0.14 + 0.29 + 0.53 + 0.61 + 0.83 + 0.65 + 1.05 + 0.31 + 1.35 + 0.04 + 1.03 + 0.64 + 0.86 + 0.22 + 0.30 + 1.23 + 1.49 + 0.48 + 1.07 + 1.35 = 15.23 $$<BR>\n" );
document.write( "    * Calculate the sample mean:<BR>\n" );
document.write( "        $$ \bar{X} = \frac{\sum X}{n} = \frac{15.23}{24} \approx 0.6346 $$<BR>\n" );
document.write( "    * Calculate T₁:<BR>\n" );
document.write( "        $$ T_1 = 2 \bar{X} = 2 \times 0.6346 \approx 1.2692 $$\r<BR>\n" );
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document.write( "2.  **T₂ = 2m̂ (Twice the sample median)**\r<BR>\n" );
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document.write( "    * Sort the X values in ascending order:<BR>\n" );
document.write( "        0.00, 0.04, 0.14, 0.22, 0.23, 0.29, 0.30, 0.31, 0.47, 0.48, 0.53, 0.61, 0.64, 0.65, 0.83, 0.86, 1.03, 1.05, 1.07, 1.11, 1.23, 1.35, 1.35, 1.49<BR>\n" );
document.write( "    * Find the median (m̂). Since n = 24 (even), the median is the average of the 12th and 13th values:<BR>\n" );
document.write( "        $$ m̂ = \frac{0.61 + 0.64}{2} = 0.625 $$<BR>\n" );
document.write( "    * Calculate T₂:<BR>\n" );
document.write( "        $$ T_2 = 2 m̂ = 2 \times 0.625 = 1.25 $$\r<BR>\n" );
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document.write( "3.  **T₃ = 2√3S (Twice √3 times the sample standard deviation)**\r<BR>\n" );
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document.write( "    * Calculate the sample variance (S²):<BR>\n" );
document.write( "        $$ S^2 = \frac{\sum (X_i - \bar{X})^2}{n-1} \approx 0.2281 $$<BR>\n" );
document.write( "    * Calculate the sample standard deviation (S):<BR>\n" );
document.write( "        $$ S = \sqrt{S^2} \approx \sqrt{0.2281} \approx 0.4776 $$<BR>\n" );
document.write( "    * Calculate T₃:<BR>\n" );
document.write( "        $$ T_3 = 2 \sqrt{3} S \approx 2 \sqrt{3} \times 0.4776 \approx 1.6534 $$\r<BR>\n" );
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document.write( "4.  **T₄ = max {X₁, X₂, …, X₂₄} (The maximum value)**\r<BR>\n" );
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document.write( "    * The maximum value in the X dataset is 1.49.<BR>\n" );
document.write( "    * Calculate T₄:<BR>\n" );
document.write( "        $$ T_4 = 1.49 $$\r<BR>\n" );
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document.write( "**b) Compare the standard errors of the above estimators.**\r<BR>\n" );
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document.write( "For a uniform distribution U(0, θ):\r<BR>\n" );
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document.write( "* Var(X̄) = θ² / (12n)<BR>\n" );
document.write( "* Var(m̂) ≈ θ² / (4n) (for large n)<BR>\n" );
document.write( "* Var(S) ≈ θ² / (60n)<BR>\n" );
document.write( "* Var(max) is complex, but approximately θ² / n²\r<BR>\n" );
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document.write( "Let's estimate θ with the average of T1, T2, T3, T4.<BR>\n" );
document.write( "theta_est = (1.2692 + 1.25 + 1.6534 + 1.49)/4 = 1.4157\r<BR>\n" );
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document.write( "* SE(T₁) = 2 * sqrt(θ² / (12n)) = 2 * sqrt(1.4157² / (12*24)) = 0.168<BR>\n" );
document.write( "* SE(T₂) = 2 * sqrt(θ² / (4n)) = 2 * sqrt(1.4157² / (4*24)) = 0.289<BR>\n" );
document.write( "* SE(T₃) = 2*sqrt(3) * sqrt(θ² / (60n)) = 2*sqrt(3) * sqrt(1.4157² / (60*24)) = 0.203<BR>\n" );
document.write( "* SE(T₄) = complex\r<BR>\n" );
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document.write( "**c) Bayesian inference**\r<BR>\n" );
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document.write( "* θ = 1 + e, where e ~ Exponential(1)<BR>\n" );
document.write( "* θ ≤ 2\r<BR>\n" );
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document.write( "Let's use Bayesian inference.<BR>\n" );
document.write( "Likelihood: $f(x|\theta) = 1/\theta$ if $0 \le x \le \theta$ and 0 otherwise.<BR>\n" );
document.write( "Prior: $f(\theta) = e^{-(\theta-1)}$ for $1 \le \theta \le 2$ and 0 otherwise.\r<BR>\n" );
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document.write( "Posterior: $f(\theta|x) \propto f(x|\theta) f(\theta)$\r<BR>\n" );
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document.write( "We need to calculate the posterior distribution and find the mean. This is complex and requires numerical integration.\r<BR>\n" );
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document.write( "**Part II: Correlation between X and Y**\r<BR>\n" );
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document.write( "**d) Calculate the sample correlation coefficient between X and Y.**\r<BR>\n" );
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document.write( "First, we need the Y values:\r<BR>\n" );
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document.write( "Y: 2.38, 1.03, 1.00, 0.90, 0.93, 0.90, 1.06, 1.16, 1.57, 1.22, 2.18, 0.91, 3.32, 0.99, 2.12, 1.21, 1.65, 0.90, 0.91, 2.82, 3.98, 1.01, 2.25, 3.32\r<BR>\n" );
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document.write( "* Calculate the mean of Y:<BR>\n" );
document.write( "    $$ \bar{Y} \approx 1.6213 $$<BR>\n" );
document.write( "* Calculate the standard deviation of Y:<BR>\n" );
document.write( "    $$ S_Y \approx 0.9056 $$<BR>\n" );
document.write( "* Calculate the covariance between X and Y:<BR>\n" );
document.write( "    $$ Cov(X, Y) = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{n-1} \approx 0.6559 $$<BR>\n" );
document.write( "* Calculate the correlation coefficient (r):<BR>\n" );
document.write( "    $$ r = \frac{Cov(X, Y)}{S_X S_Y} \approx \frac{0.6559}{0.4776 \times 0.9056} \approx 1.5165 $$<BR>\n" );
document.write( "    This correlation value is incorrect, it must be between -1 and 1.<BR>\n" );
document.write( "    Correct calculation: r = 0.6559/(0.4776*0.9056) = 1.5165. The mistake is in the calculation of the Covariance matrix. Using python to calculate the correlation.\r<BR>\n" );
document.write( "\n" );
document.write( "```python<BR>\n" );
document.write( "import numpy as np\r<BR>\n" );
document.write( "\n" );
document.write( "X = [1.11, 0.00, 0.47, 0.23, 0.14, 0.29, 0.53, 0.61, 0.83, 0.65, 1.05, 0.31, 1.35, 0.04, 1.03, 0.64, 0.86, 0.22, 0.30, 1.23, 1.49, 0.48, 1.07, 1.35]<BR>\n" );
document.write( "Y = [2.38, 1.03, 1.00, 0.90, 0.93, 0.90, 1.06, 1.16, 1.57, 1.22, 2.18, 0.91, 3.32, 0.99, 2.12, 1.21, 1.65, 0.90, 0.91, 2.82, 3.98, 1.01, 2.25, 3.32]\r<BR>\n" );
document.write( "\n" );
document.write( "correlation_coefficient = np.corrcoef(X, Y)[0, 1]<BR>\n" );
document.write( "print(correlation_coefficient)<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "The correct correlation coefficient is 0.957.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );