Question

The accompanying data on x= head circumference z score (a comparison score with peers of the same age - a positive score suggests a larger size than for peers) at age of 6 to 14 months and y =volume of cerebral grey matter (in ml) at age 2 to 5 years were read from a graph in an article.
A) what is the value of the correlation coefficient? (Give the answer to two decimal places.) r=
B)Find the equation of the least- squares lines. (Give the answer to two decimal places.)
C) Predict the volume of cerebral grey matter for a child whose head circumference z score at age 12 months was 1.9. (Give the answer to two decimal places.)

Answer 1

Answer:

$r=0.77$, $\hat{y}=770.50+39.37x$ and volume of cerebral brey matter for a child whose head circumference Z score at age 12 months was 1.9 is  $\hat {y}=920.10$

Step-by-step explanation:

We attached a table below,

$(a)$  We attached a table below,

Therefore ,

correlation coefficient  $r=\frac{\frac{1}{18}(21766.7)-(\frac{27.25}{18} )(\frac{13890}{18} ) }{\sqrt{\frac{1}{18} (59.8435)-(\frac{27.25}{18} )^2}\sqrt{\frac{1}{18}(10767400)-(\frac{13890}{18} )^2 } }$

                                         $=\frac{41.04}{53.19}$

                                         $=0.77$

$(b)$    We calculate the least squares equation.

      $b_1=r(\frac{\sigma_y}{\sigma_x} )\\ =(0.77)(\frac{\sqrt{\frac{1}{18} (10767400)-(\frac{13890}{18} )^2} }{\sqrt{\frac{1}{18}(59.8435)-(\frac{27.25}{18} )^2 } } )\\ =0.77(\frac{52.15}{1.02} )\\ =0.77(51.13)\\ =39.37$

Now , calculate intercept coefficient.

$b_0=\bar{y}-r\bar{x}$

    $=(\frac{\sum{y}}{n} )-0.77(\frac{\sum{x}}{n} )\\=(\frac{13890}{18} )-0.77(\frac{27.25}{18} )\\=771.67-1.17\\=770.50$

Therefore, the regression equation is, $\hat{y}=770.50+39.37x$

$(c)$    Find the volume of cerebral brey matter for a child whose head circumference Z score at age 12 months was 1.8

$\hat{y}=770.50+39.37x$

   $=770.50+39.37(1.9)\\=770.50+74.80\\=920.10$