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INSTANT ANSWER

5. Given that \( \mathrm{SSE}=\sum_{i=1}^{n}\left(y_{i}-\hat{\beta}_{0}-\hat{\beta}_{1} x_{i}\right)^{2} \), towards deriving \( \mathrm{SSE}=S_{y y}-\hat{\beta}_{1} S_{x y} \), \( \mathrm{SSE} \) can be expressed as A. \( -2 \sum_{i=1}^{n}\left[y_{i}-\left(\hat{\beta}_{0}+\hat{\beta}_{1} x_{i}\right)\right] \) B. \( -2 \sum_{i=1}^{n}\left[y_{i}-\left(\hat{\beta}_{0}+\hat{\beta}_{1}\right) x_{i}\right] \) C. \( \sum_{i=1}^{n}\left(y_{i}-\bar{y}+\hat{\beta}_{1} \bar{x}-\hat{\beta}_{1} x_{i}\right)^{2} \) D. \( \sum_{i=1}^{n}\left(y_{i}^{2}+n \hat{\beta_{0}}-\hat{\beta}_{1} x_{i}^{2}\right)^{2} \) 6. The SSE can then be arranged as A. \( \sum_{i=1}^{n}\left[\left(y_{i}-\bar{y}\right)-\hat{\beta}_{1}\left(x_{i}-\bar{x}\right)\right]^{2} \) B. \( \sum_{i=1}^{n}\left[\left(y_{i}-\bar{y}\right)-\left(\hat{\beta}_{0}+\hat{\beta}_{1}\right)\left(x_{i}-\bar{x}\right)\right]^{2} \) C. \( \sum_{i=1}^{n}\left[y_{i}^{2}-\left(n \hat{\beta_{0}}+\hat{\beta}_{1}\right) x_{i}^{2}\right] \) D. \( \sum_{i=1}^{n} y_{i}-n \hat{\beta}_{0}+\hat{\beta}_{1} \sum_{i=1}^{n} x_{i} \)

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INSTANT ANSWER

Consider the simple linear regression model \[ Y_{i}=\beta_{0}+\beta_{1} x_{i}+\epsilon_{i} \quad i=1,2, \ldots, n \] where the \( \epsilon_{i} \) 's are independent and identically distributed random variables with \( E\left(\epsilon_{i}\right)=0 \). The Sum Squared Error (SSE) is given by \[ \mathrm{SSE}=\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}=\sum_{i=1}^{n}\left(y_{i}-\hat{\beta}_{0}-\hat{\beta}_{1} x_{i}\right)^{2} \] where \( \hat{y}_{i} \) is the predicted value of \( y \) when \( x=x_{i} \). - Derive the following identity: \[ \mathrm{SSE}=S_{y y}-\hat{\beta}_{1} S_{x y} \] where \[ \begin{aligned} S_{y y} & =\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2} \\ S_{x y} & =\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right) \end{aligned} \] - Consider SSE and \( S_{y y} \), which of the following is correct? \[ \mathrm{SSE} \leq S_{y y} \] or \[ \mathrm{SSE} \geq S_{y y} \] Write down your proof. [Hint: \( \hat{\beta}_{1}=S_{x y} / S_{x x} \); use the above identity]

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INSTANT ANSWER

Suppose that we have postulated the model \[ Y_{i}=\beta_{1} x_{i}+\epsilon_{i} \quad i=1,2, \ldots, n, \] where the \( \epsilon_{i} \) 's are independent and identically distributed random variables with \( E\left(\epsilon_{i}\right)=0 \) and \( \operatorname{Var}\left(\epsilon_{i}\right)=\sigma^{2} \). Then \( \hat{y_{i}}=\hat{\beta}_{1} x_{i} \) is the predicted value of \( y \) when \( x=x_{i} \) and \( \mathrm{SSE}= \) \( \sum_{i=1}^{n}\left[y_{i}-\hat{\beta}_{1} x_{i}\right]^{2} \) - Derive \( \hat{\beta}_{1} \), the least-squares estimator of the parameter \( \beta_{1} \). - Show if \( \hat{\beta}_{1} \) is or is not an unbiased estimator of the parameter \( \beta_{1} \). - Find the variance of \( \hat{\beta}_{1} \).

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ANSWERED

Kirsty Gledhill

Numerade educator

Let ( X_{1}, X_{2}, ldots, X_{n} ) be a random sample from a normal distribution with mean ( mu ) and variance ( sigma^{2} ). 1. Derive the method-of-moments estimators of ( mu ) and ( sigma^{2} ); 2. Show that if the method-of-moments estimators of ( mu ) and ( sigma^{2} ) are unbiased estimators or not.

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ANSWERED

Shu Naito

Numerade educator

A manufacturing company produces bags of flour for commercial use. It is known that the variation in weight of bags produced by this process has a normal distribution about its target weight with a standard deviation of 1 kg. 1. What should the manufacturing company set as its target weight in order for 97.5% of its bags to have a weight of at least 100 kg? 2. A random sample of 20 bags was weighed with the following results in kg. 101.26 99.76 98.96 100.36 98.94 98.82 98.51 100.97 101.04 100.31 98.24 100.58 100.20 99.28 99.04 99.95 99.34 99.54 97.89 97.98 Determine a 95% confidence interval for process's actual average bag weight 3. A government inspector wishes to investigate the average weight of bags being produced by this manufacturing company. If a 95% confidence interval for the process's actual average bag weight of width no more than 0.2 kg is required, what is the minimum number of bags that need to be sampled by the government inspector?

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Pranathi C.