the following table shows students test score...
the following table shows students test scores on the first two tests in an introductory chemistry class.\nchemistry test scores\nfirst test, x: 58 62 47 91 51 68 61 95 92 42 53 61\nsecond test, y: 53 58 52 83 50 56 52 83 81 42 51 56\ncopy data\nstep 1 of 2: find an equation of the least - squares regression line. round your answer to three decimal places, if necessary.\nanswer\ny^ = + x
Answer
# Explanation:
## Step1: Calculate the means of \(x\) and \(y\)
Let \(n = 12\).
\(\bar{x}=\frac{58 + 62+47+91+51+68+61+95+92+42+53+61}{12}=\frac{781}{12}\approx65.083\)
\(\bar{y}=\frac{53 + 58+52+83+50+56+52+83+81+42+51+56}{12}=\frac{687}{12}= 57.25\)
## Step2: Calculate the numerator and denominator for the slope \(b_1\)
The formula for \(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=(58 - 65.083)(53-57.25)+(62 - 65.083)(58 - 57.25)+\cdots+(61-65.083)(56 - 57.25)\)
\(=( - 7.083)\times(-4.25)+(-3.083)\times0.75+\cdots+(-4.083)\times(-1.25)\)
\(=30.10275-2.31225+\cdots + 5.10375\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=477.25\)
\(\sum_{i=1}^{n}(x_i-\bar{x})^2=(58 - 65.083)^2+(62 - 65.083)^2+\cdots+(61 - 65.083)^2\)
\(=(-7.083)^2+(-3.083)^2+\cdots+(-4.083)^2\)
\(\sum_{i=1}^{n}(x_i-\bar{x})^2 = 1976.917\)
\(b_1=\frac{477.25}{1976.917}\approx0.241\)
## Step3: Calculate the y - intercept \(b_0\)
The formula for \(b_0=\bar{y}-b_1\bar{x}\)
\(b_0 = 57.25-0.241\times65.083\)
\(b_0=57.25 - 15.685\)
\(b_0 = 41.565\)
# Answer:
\(\hat{y}=41.565 + 0.241x\)