91Ó°ÊÓ

Given,

\(x=3\)

Computation of table:

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

\(=0-(15)(5)/5\)

\(=0-75/5\)

\(=0-15\)

\(=-15\)

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

\(=69-(15)^{2}/5\)

\(=69-225/5\)

\(=69-45\)

\(=24\)

The total sum of squares SST is given by,

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

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

\(=25-25/5\)

\(=25-5\)

\(=20\)

The regression sum of squares SSR is given by,

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

\(=\frac{(-15)^{2}}{24}=\frac{225}{24}=9.375\)

\(SSE=SST-SSR\)

\(=20-9.375\)

\(=10.625\)

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

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

\(=\sqrt{\frac{10.625}{5-2}}\)

\(=1.881931632\)

\(\approx 1.881\)

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

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

\(=\frac{-15}{24}\)

\(=-0.625\)

The formula for calculating the value of y-intercept is

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

\(=\frac{1}{5}(5+0.625(15))\)

\(=\frac{1}{5}(14.375)\)

\(=2.875\)

So, the regression equation is \(\hat{y_{p}}=2.875-0.625x_{p}\)

The formula for calculating the value of the point estimate is obtained by substituting the value of \(x_{p}=3\) in the regression equation.

\(\hat{y_{p}}=12.875-0.625x_{p}\)

\(=2.875+0.625(3)\)

\(=1\)

The point estimate is \(\hat{y_{p}}=1\)

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

\(df=n-2\)

\(=5-2\)

\(=3\)

From technology, \(t_{\alpha/2}=t_{0.05/2}=t_{0.025}=3.187\)

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, \(x_{p}=3\),

\(\hat{y_{p}} =1\),

\(s_{e}=1.882\),

\(S_{xx}=24\).

So, \(1\pm 3.187\times (1.882) \sqrt{\frac{1}{5}+\frac{(3-15/5)^{2}}{24}}\)

\(1\pm 5.988524 \sqrt{0.2}\)

Or \(1\pm 2.67814935\)

Or \(-1.68\) to \(3.68\)

Therefore, the \(95%\) confidence interval for the conditional mean is \(-1.68\) to \(3.68\).

The regression equation is \(\hat{y_{p}}=2.875-0.625x_{p}\)

The predicted value is obtained by substituting the value of \(x_{p}=3\) in the regression equation.

\(\hat{y_{p}}=2.875-0.625x_{p}\)

\(=2.875+0.625(3)\)

\(=1\)

The predicted value is \(\hat{y_{p}}= 1\)

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

\(df=n-2\)

\(=5-2\)

\(=3\)

From technology, \(t_{\alpha/2}=t_{0.05/2}=t_{0.025}=3.187\)

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, \(x_{p}=3\),

\(\hat{y_{p}} =1\),

\(s_{e}=1.882\),

\(S_{xx}=24\).

So, \(1\pm 3.187\times (1.882) \sqrt{\frac{1}{5}+\frac{(3-15/5)^{2}}{24}}\)

\(1\pm 5.988524 \sqrt{1+0.2}\)

Or \(1\pm 6.560099362\)

Or \(-5.56\) to \(7.56\)

Therefore, the \(95%\) confidence interval for the conditional mean is \(-5.56 \) to \(7.56\).