EasyMathematicsHigh School (11–12)Statistics and ProbabilityNumerical

A school gives a practice test and then a real test…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A school gives a practice test and then a real test. For a large group, the practice scores have a mean of $40$ and a standard deviation of $10$ . The real scores have a mean of $65$ and a standard deviation of $8$ . The correlation between the two is $0.75$ . The line of the real score $y$ on the practice score $x$ is wanted. Find its slope $b_{yx}$ and its intercept. Then find the predicted real score for a student who scored $55$ in the practice.

Need a reminder? The key ideaShow

$b_{yx} = \dfrac{S_{xy}}{S_{xx}} = r\dfrac{\sigma_y}{\sigma_x}$ and $b_{xy} = \dfrac{S_{xy}}{S_{yy}} = r\dfrac{\sigma_x}{\sigma_y}$ . The line of $y$ on $x$ is $y - \bar{y} = b_{yx}(x - \bar{x})$ . Both lines pass through $(\bar{x}, \bar{y})$ , and $b_{yx}\,b_{xy} = r^2$ .

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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