14. Correlation and Regression

14.7 Confidence Interval for y′ at a Given x

At a fixed x (that is important to remember) the confidence interval for y is

    \[ y^{\prime} - E < y < y^{\prime} + E \]

where

    \[ E = t_{\cal{C}} \; s_{\mbox{est}} \sqrt{1+ \frac{1}{n} + \frac{n(x - \overline{x})^{2}}{n(\sum x^{2})-(\sum x)^{2}}} \]

where, as usual, t_{\cal{C}} comes from the t Distribution Table with \nu = n-2.

Example 14.5 : Continuing from Example 14.4 (so you can see how an exam will go), say we want to predict the grade (y) in terms of a 95\% confidence interval for the number of absences (x) equal to 10.

First, find the value predicted from the regression line, which we previously found to be :

    \[ y^{\prime} = 102.493 - 3.622 x \]

at x = 10. The result is

    \[ y^{\prime} = 102.493 - 3.622 (10) = 66.273 \]

Furthermore, from the last example, we found

    \[ s_{\mbox{est}} = 6.06 \]

and, from the completed data table (Example 14.3)

    \[ \sum x = 57 \;\;\; \sum x^{2} = 579 \]

We still need t{\cal{C}} and \overline{x}. Using our sums:

    \[ \overline{x} = \frac{\sum x}{n} = \frac{57}{7} = 8.143 \]

and from t Distribution Table for the 95\% confidence interval, \nu = 7-2 = 5 we get

    \[ t{\cal{C}} = 2.571 \]

Now we compute E :

    \begin{eqnarray*} E & = & t_{\cal{C}} \; s_{\mbox{est}} \sqrt{1+ \frac{1}{n} + \frac{n(x - \overline{x})^{2}}{n(\sum x^{2})-(\sum x)^{2}}}\\ E & = & (2.571) \; (6.06) \sqrt{1+ \frac{1}{7} + \frac{7(10 - 8.143)^{2}}{7(579)-(52)^{2}}}\\ E & = & 15.58026 \sqrt{1 + 0.1428571 + \frac{24.139}{804}}\\ E & = & 16.77 \end{eqnarray*}

So

    \begin{eqnarray*} y^{\prime} - E & < y < & y^{\prime} + E \\ 66.273 - 16.77 & < y < & 66.273 + 16.77 \\ 49.5 & < y < & 83.0 \end{eqnarray*}

This is the 95\% confidence interval for predicting the mark of a person who was absent for 10 days.

Important: s_{\mbox{set}} is independent of x but E is not. So confidence intervals look like :

The reason for this variance of the width of the confidence interval comes from the uncertainty in the slope b. You can make plots like the one above in SPSS.

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