A researcher investigates the relationship between the number of charge cycles, nnn, and the maximum flight duration, ddd minutes, of a drone's battery. A random sample of 30 batteries is analyzed and the results are recorded.
The data were coded using x=n−405x = \frac{n - 40}{5}x=5n−40 and y=d−102y = \frac{d - 10}{2}y=2d−10 and the following summary statistics were obtained:
∑x=60,∑y=150,∑x2=380,∑y2=900,∑xy=180 \sum x = 60, \quad \sum y = 150, \quad \sum x^2 = 380, \quad \sum y^2 = 900, \quad \sum xy = 180 ∑x=60,∑y=150,∑x2=380,∑y2=900,∑xy=180Calculate the values of SxyS_{xy}Sxy and 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 ddd on nnn is given by d=k+mnd = k + mnd=k+mn.
Show that m=−0.185m = -0.185m=−0.185 to 3 significant figures and find the value of kkk.
Estimate the flight duration of a battery that has undergone 100 charge cycles.
Interpret the value of mmm in the context of this study.
The researcher plans to use a battery for an additional 25 charge cycles beyond its current usage. Estimate the change in flight duration for this period based on the model.
