Question

find the equation for the least squares regression line of the data described below. a medical student reads that with enough sun exposure, the human body can manufacture vitamin d3. this sounds too amazing to be true, so he decides to test it out. over the next few weeks, he spends a varying number of hours in the sun, x. at the end of each week, he measures the concentration of vitamin d3 in his blood (in nanograms per milliliter), y. round your answers to the nearest thousandth. y = x +

Explanation:

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\).
For \(x\) - values \(3,4,5,6,8,10\), \(\sum x=3 + 4+5+6+8+10 = 36\), \(n = 6\), so \(\bar{x}=\frac{36}{6}=6\).
For \(y\) - values \(22,11,34,19,44,57\), \(\sum y=22 + 11+34+19+44+57=187\), \(\bar{y}=\frac{187}{6}\approx31.167\).

Step2: Calculate the numerator and denominator for the slope \(b\)

The formula for \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\).
\((x_{1}-\bar{x})(y_{1}-\bar{y})=(3 - 6)(22-31.167)=(-3)(-9.167) = 27.501\)
\((x_{2}-\bar{x})(y_{2}-\bar{y})=(4 - 6)(11 - 31.167)=(-2)(-20.167)=40.334\)
\((x_{3}-\bar{x})(y_{3}-\bar{y})=(5 - 6)(34 - 31.167)=(-1)(2.833)=-2.833\)
\((x_{4}-\bar{x})(y_{4}-\bar{y})=(6 - 6)(19 - 31.167)=0\times(-12.167)=0\)
\((x_{5}-\bar{x})(y_{5}-\bar{y})=(8 - 6)(44 - 31.167)=2\times12.833 = 25.666\)
\((x_{6}-\bar{x})(y_{6}-\bar{y})=(10 - 6)(57 - 31.167)=4\times25.833=103.332\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=27.501+40.334-2.833 + 0+25.666+103.332=193.999\)

\((x_{1}-\bar{x})^{2}=(3 - 6)^{2}=9\)
\((x_{2}-\bar{x})^{2}=(4 - 6)^{2}=4\)
\((x_{3}-\bar{x})^{2}=(5 - 6)^{2}=1\)
\((x_{4}-\bar{x})^{2}=(6 - 6)^{2}=0\)
\((x_{5}-\bar{x})^{2}=(8 - 6)^{2}=4\)
\((x_{6}-\bar{x})^{2}=(10 - 6)^{2}=16\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=9 + 4+1+0+4+16=34\)

So \(b=\frac{193.999}{34}\approx5.706\)

Step3: Calculate the intercept \(a\)

The formula for \(a=\bar{y}-b\bar{x}\).
\(a = 31.167-5.706\times6=31.167 - 34.236=-3.069\)

Answer:

\(y = 5.706x-3.069\)