A marine biologist is investigating the relationship between the time a coral specimen has been submerged in a controlled nursery and its total vertical height. She monitors a random sample of 30 specimens and records the time submerged, ttt months, and the vertical height, hhh cm.
The data were coded using x=t−104x = \frac{t - 10}{4}x=4t−10 and y=h5y = \frac{h}{5}y=5h and the following statistics were obtained:
∑x=150,∑y=360,∑x2=1150,∑y2=4670,∑xy=2080 \sum x = 150, \quad \sum y = 360, \quad \sum x^2 = 1150, \quad \sum y^2 = 4670, \quad \sum xy = 2080 ∑x=150,∑y=360,∑x2=1150,∑y2=4670,∑xy=2080Find the value of SxyS_{xy}Sxy and the value of SxxS_{xx}Sxx.
Find the equation of the least squares regression line of yyy on xxx in the form y=α+βxy = \alpha + \beta xy=α+βx.
The least squares regression line of hhh on ttt is h=k+mth = k + mth=k+mt.
Show that m=0.875m = 0.875m=0.875 and find the value of kkk.
Estimate the vertical height of a coral specimen that has been submerged for 24 months.
Interpret the value of mmm in context.
The biologist intends to leave a specimen submerged for an additional 8 months. Estimate the total increase in vertical height for this period based on the model.
