Question

data are recorded for a random sample of 25 adults. the table below shows summary statistics for the number of children in their family and the total charges billed by health insurance (in dollars) for these children. the correlation between number of children and medical costs was found to be ( r=-0.26 ).
which of the following is the approximate equation of the least - squares regression line for predicting ( y = ) charges based on ( x = ) number of children?
a) ( hat{y}=24,918 - 2x )
b) ( hat{y}=14,618 - 3.374x )
c) ( hat{y}=3.374 - 14,618x )
d) ( hat{y}=-4,154 + 12,977x )
e) ( hat{y}=10,299 - 3,374x )

Explanation:

Step1: Calculate the slope \(b\)

The formula for the slope \(b\) of the least - squares regression line is \(b = r\frac{s_{y}}{s_{x}}\).
Given \(r=- 0.26\), \(s_{y}=12354.05\), \(s_{x}=0.952\).
\(b=-0.26\times\frac{12354.05}{0.952}\approx - 3374\)

Step2: Calculate the intercept \(a\)

The formula for the intercept \(a\) is \(a=\bar{y}-b\bar{x}\).
Given \(\bar{y} = 12458.79\), \(\bar{x}=0.64\), \(b\approx - 3374\)
\(a = 12458.79-(-3374)\times0.64=12458.79 + 2159.36=14618.15\approx14618\)

Answer:

B. \(\hat{y}=14618 - 3.374x\)