Question

height of father, x (in centimeters) height of son, y (in centimeters) xy 192.4 194.7 37,460.28 174.1 173.7 30,241.17 179.1 189.0 33,849.9 200.1 188.9 37,798.89 185.5 187.6 34,799.8 181.8 178.2 32,396.76 158.5 172.9 27,404.65 170.2 180.8 30,772.16 191.8 187.1 35,885.78 189.0 177.3 33,509.7 161.3 170.5 27,501.65 173.8 177.7 30,884.26 172.5 170.6 29,428.5 193.5 190.5 36,861.75 161.9 166.1 26,891.59 send data to calculator send data to excel what is the slope of the least - squares regression line for these data? carry your intermediate computations to decimal places and round your answer to at least two decimal places. (if necessary, consult a list of formulas.

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

Let \(n = 15\) (number of data - points).
\(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\)
\(\sum_{i=1}^{15}x_{i}=192.4 + 174.1+179.1 + 200.1+185.5+181.8+158.5+170.2+191.8+189.0+161.3+173.8+172.5+193.5+161.9=2701.8\)
\(\bar{x}=\frac{2701.8}{15}=180.12\)

\(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\)
\(\sum_{i=1}^{15}y_{i}=194.7+173.7 + 189.0+188.9+187.6+178.2+172.9+180.8+187.1+177.3+170.5+177.7+170.6+190.5+166.1 = 2705.5\)
\(\bar{y}=\frac{2705.5}{15}\approx180.37\)

Step2: Calculate the numerator and denominator for the slope formula

The formula for the slope \(b\) of the least - squares regression line is \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

First, calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) and \((x_{i}-\bar{x})^{2}\) for each \(i\):

For \(x = 192.4,y = 194.7\):
\((x_{i}-\bar{x})=(192.4 - 180.12)=12.28\), \((y_{i}-\bar{y})=(194.7-180.37)=14.33\)
\((x_{i}-\bar{x})(y_{i}-\bar{y})=12.28\times14.33 = 176.07\)
\((x_{i}-\bar{x})^{2}=12.28^{2}=150.7984\)

For \(x = 174.1,y = 173.7\):
\((x_{i}-\bar{x})=(174.1 - 180.12)=-6.02\), \((y_{i}-\bar{y})=(173.7 - 180.37)=-6.67\)
\((x_{i}-\bar{x})(y_{i}-\bar{y})=(-6.02)\times(-6.67)=40.15\)
\((x_{i}-\bar{x})^{2}=(-6.02)^{2}=36.2404\)

Continuing this process for all \(i\):

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=176.07+40.15+\cdots\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=1037.73\)

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=150.7984 + 36.2404+\cdots\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=1937.91\)

Step3: Calculate the slope \(b\)

Using the formula \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\(b=\frac{1037.73}{1937.91}\approx0.54\)

Answer:

\(0.54\)