The accompanying data on x = head circumference z score (a comparison score with

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 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.

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 6 to 14 months and y volume of cerebral grey matter (in m at age 2 to 5 years were read from a graph in an article Head Circumfer- Cerebral Grey Matter (ml) 2-5 yr encez Scores at 6-14 Months 680 690 700 720 740 740 750 750 760 780 790 810 815 820 825 835 840 845 0.70 -0.25 0.30 0.35 1.5 1.1 2.0 1.1 2.0 2.1 2.8 2.2 0.95 2.35 2.3 2.2 (a) What is the value of the correlation coefficient? (Give the answer to two decimal places.) (b) Find the equation of the least-squares line. (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 1.8. (Give the answer to two decimal places.)

Explanation / Answer

we will evaluate that if there is a linear relationship possible or not.

let say linear line equation is y = a + bx

where y = volume of cerebral grey matter (in ml) at age 2 to 5 years

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 6 to 14 month

so The table which will help us to calculate linear regression line is

a = [(y) (x2 ) - (x) (xy)]/ [ n (x2 ) - (x)2 ] FIrst we will caluate part b, the linear line equation

a = [13890 * 49.73 - 24.6 * 19685] / [18 * 49.73 - 24.62]

a = 206498.7/ 289.98 = 712.11

b =  [ n(xy) - (x)((y)]/ [ n (x2 ) - (x)2 ]

b = [18 * 19685 - 24.6 * 13890] / [ 18 * 49.73 - 24.62 ]

b = 12636/ 289.98 = 43.57

so line equation y^ = 712.11 + 43.58x

(a) Correaltion coefficient r = [n(xy) - (x)((y)] / sqrt [ (n (x2 ) - (x)2) ( n (y2 ) - (y)2

r = [ 18 * 19685 - 24.6 * 13890] / sqrt [289.98 * 881100]

r = 12636/ 15984.41 = 0.79

(c) Z score here is = 1.8

so by regression line equation

cerebral grey matter = 712.11 + 43.58 * 1.8 = 790.55 ml

Head Circumference z scores (x) Cerebral Grey Matter (y) x^2 xy y^2 -0.7 680 0.49 -476 462400 -0.25 700 0.0625 -175 490000 0.3 720 0.09 216 518400 0.35 740 0.1225 259 547600 0.95 825 0.9025 783.75 680625 1.1 750 1.21 825 562500 1.1 760 1.21 836 577600 1.1 780 1.21 858 608400 1.2 690 1.44 828 476100 1.5 740 2.25 1110 547600 2 750 4 1500 562500 2 790 4 1580 624100 2.1 810 4.41 1701 656100 2.2 820 4.84 1804 672400 2.2 845 4.84 1859 714025 2.3 840 5.29 1932 705600 2.35 835 5.5225 1962.25 697225 2.8 815 7.84 2282 664225 sum 24.6 13890 49.73 19685 10767400

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