Question

The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults. What is the best predicted value for y given x = 41? Assume that the variables x and y have a significant correlation.

Answer 1

Answer:

$\sum_{i=1}^n x_i =459$

$\sum_{i=1}^n y_i =1227$

$\sum_{i=1}^n x^2_i =24059$

$\sum_{i=1}^n y^2_i =168843$

$\sum_{i=1}^n x_i y_i =63544$

With these we can find the sums:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=24059-\frac{459^2}{9}=650$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=63544-\frac{459*1227}{9}=967$

And the slope would be:

$m=\frac{967}{650}=1.488$

Nowe we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{459}{9}=51$

$\bar y= \frac{\sum y_i}{n}=\frac{1227}{9}=136.33$

And we can find the intercept using this:

$b=\bar y -m \bar x=136.33-(1.488*51)=60.442$

So the line would be given by:

$y=1.488 x +60.442$

And then the best predicted value of y for x = 41 is:

$y=1.488*41 +60.442 =121.45$

Step-by-step explanation:

For this case we assume the following dataset given:

x: 38,41,45,48,51,53,57,61,65

y: 116,120,123,131,142,145,148,150,152

For this case we need to calculate the slope with the following formula:

$m=\frac{S_{xy}}{S_{xx}}$

Where:

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$

So we can find the sums like this:

$\sum_{i=1}^n x_i =459$

$\sum_{i=1}^n y_i =1227$

$\sum_{i=1}^n x^2_i =24059$

$\sum_{i=1}^n y^2_i =168843$

$\sum_{i=1}^n x_i y_i =63544$

With these we can find the sums:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=24059-\frac{459^2}{9}=650$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=63544-\frac{459*1227}{9}=967$

And the slope would be:

$m=\frac{967}{650}=1.488$

Nowe we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{459}{9}=51$

$\bar y= \frac{\sum y_i}{n}=\frac{1227}{9}=136.33$

And we can find the intercept using this:

$b=\bar y -m \bar x=136.33-(1.488*51)=60.442$

So the line would be given by:

$y=1.488 x +60.442$

And then the best predicted value of y for x = 41 is:

$y=1.488*41 +60.442 =121.45$