91Ó°ÊÓ

Given,

\(x=1\)

Computation table:

\(S_{xy}=\sum x_{i}y_{i}-(\sum x_{i})(\sum y_{i})/n\)

\(=70-(10)(25)/5\)

\(=70-250/5\)

\(=70-50\)

\(=20\)

\(S_{xx}=\sum x^{2}_{i}-(\sum x_{i})^{2}/n\)

\(=30-(10)^{2}/5\)

\(=30-100/5\)

\(=30-20\)

\(=10\)

The total sum of squares SST is given by,

\(S_{yy}=\sum y^{2}_{i}-(\sum y_{i})^{2}/n\)

\(=171-(25)^{2}/5\)

\(=171-625/5\)

\(=171-125\)

\(=46\)

The regression sum of squares SSR is given by,

\(SSR=\frac{S_{xy}^{2}}{S_{xx}}\)

\(=\frac{(20)^{2}}{10}=\frac{400}{10}=40\)

\(SSE=SST-SSR\)

\(=46-40\)

\(=6\)

The formula for calculating the standard error of the estimate is,

\(s_{e}=\sqrt{\frac{SSE}{n-2}}\)

\(=\sqrt{\frac{6}{5-2}}\)

\(=1.414213562\)

\(\approx 1.4142\)

The formula for calculating the slope of the regression line is.

\(b_{1}=\frac{S_{xy}}{S_{xx}}\)

\(=\frac{20}{10}\)

\(=2\)

The formula for calculating the value of y-intercept is

\(b_{0}=\frac{1}{n}(\sum y_{i}-b\sum x_{i})\)

\(=\frac{1}{5}(25-2(10))\)

\(=\frac{1}{5}(5)\)

\(=1\)

So, the regression equation is \(\hat{y_{p}}=1+2x_{p}\)

The formula for calculating the value of the point estimate is obtained by substituting the value of \(x_{p}=2\) in the regression equation.

\(\hat{y_{p}}=1+2x_{p}\)

\(=1+2(1)\)

\(=3\)

The point estimate is \(\hat{y_{p}}=-3\)

STEP 1: For a \(95%\) confidence interval, \(\alpha=0.05\). Because \(n=5\),

\(df=n-2\)

\(=5-2\)

\(=3\)

From technology, \(t_{\alpha/2}=t_{0.05/2}=t_{0.025}=3.182\)

STEP 2:

The formula for calculating the end points of the confidence interval for the conditional mean of the response variable are

\(\hat{y_{p}}\pm t_{\alpha/2}\times s_{e}\sqrt{\frac{1}{n}+\frac{(x_{p}-\sum x_{i}/n)^{2}}{S_{xx}}}\)

We have, \(x_{p}=1\),

\(\hat{y_{p}} =3\),

\(s_{e}=1.4142\),

\(S_{xx}=10\).

So, \(-3\pm 3.182\times (1.4142) \sqrt{\frac{1}{5}+\frac{(1-10/5)^{2}}{10}}\)

\(-3\pm 4.4999844 \sqrt{0.5+0.2}\)

Or \(-3\pm 2.464742964\)

Or \(0.535\) to \(5.465\)

Therefore, the \(95%\) confidence interval for the conditional mean is \(0.535\) to \(5.465 \).

The regression equation is \(\hat{y_{p}}=1+2x_{p}\)

The predicted value is obtained by substituting the value of \(x_{p}=1\) in the regression equation.

\(\hat{y_{p}}=1+2x_{p}\)

\(=1+2(1)\)

\(=3\)

The predicted value is \(\hat{y_{p}}= 3\)

STEP 1: For a \(95%\) confidence interval, \(\alpha=0.05\). Because \(n=5\),

\(df=n-2\)

\(=5-2\)

\(=3\)

From technology, \(t_{\alpha/2}=t_{0.05/2}=t_{0.025}=3.182\)

STEP 2:

The formula for calculating the end points of the confidence interval for the conditional mean of the response variable are

\(\hat{y_{p}}\pm t_{\alpha/2}\times s_{e}\sqrt{\frac{1}{n}+\frac{(x_{p}-\sum x_{i}/n)^{2}}{S_{xx}}}\)

We have, \(x_{p}=1\),

\(\hat{y_{p}} =3\),

\(s_{e}=1.4142\),

\(S_{xx}=10\).

So, \(-3\pm 3.182\times (1.4142) \sqrt{\frac{1}{5}+\frac{(1-10/5)^{2}}{10}}\)

\(-3\pm 4.4999844 \sqrt{1+0.3}\)

Or \(-3\pm 5.130771626\)

Or \(-2.132\) to \(8.132\)

Therefore, the \(95%\) prediction interval is \(-2.132\) to \(8.132\)