10) two separate tests are designed to measure a students ability to solve problems. several students are…

Explanation:

Step1: Calculate means

Let $x = [2,4,5,6]$, $y=[7,11,13,20]$. $\bar{x}=\frac{2 + 4+5+6}{4}=4.25$, $\bar{y}=\frac{7 + 11+13+20}{4}=12.75$.

Step2: Calculate slope $b$

$b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$. $\sum_{i = 1}^{4}(x_i-\bar{x})(y_i - \bar{y})=(2 - 4.25)(7-12.75)+(4 - 4.25)(11 - 12.75)+(5 - 4.25)(13 - 12.75)+(6 - 4.25)(20 - 12.75)$ $=(- 2.25)\times(-5.75)+(-0.25)\times(-1.75)+0.75\times0.25 + 1.75\times7.25$ $=12.9375+0.4375 + 0.1875+12.6875=26.25$. $\sum_{i=1}^{4}(x_i-\bar{x})^2=(2 - 4.25)^2+(4 - 4.25)^2+(5 - 4.25)^2+(6 - 4.25)^2$ $=(-2.25)^2+(-0.25)^2+0.75^2+1.75^2$ $=5.0625 + 0.0625+0.5625+3.0625 = 8.75$. $b=\frac{26.25}{8.75}=3$.

Step3: Calculate intercept $a$

$a=\bar{y}-b\bar{x}=12.75-3\times4.25 = 0$. The regression - line equation is $\hat{y}=3.0x$.

Answer:

D. $\hat{y}=3.0x$

Explanation:

Step1: Calculate means

Let $x=[0,3,4,5,12]$, $\bar{x}=\frac{0 + 3+4+5+12}{5}=4.8$, $y = [8,2,6,9,12]$, $\bar{y}=\frac{8 + 2+6+9+12}{5}=7.4$.

Step2: Calculate slope $b$

$b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$. $\sum_{i=1}^{5}(x_i - \bar{x})(y_i-\bar{y})=(0 - 4.8)(8 - 7.4)+(3 - 4.8)(2 - 7.4)+(4 - 4.8)(6 - 7.4)+(5 - 4.8)(9 - 7.4)+(12 - 4.8)(12 - 7.4)$ $=(-4.8)\times0.6+(-1.8)\times(-5.4)+(-0.8)\times(-1.4)+0.2\times1.6+7.2\times4.6$ $=-2.88 + 9.72+1.12+0.32+33.12=41.4$. $\sum_{i=1}^{5}(x_i-\bar{x})^2=(0 - 4.8)^2+(3 - 4.8)^2+(4 - 4.8)^2+(5 - 4.8)^2+(12 - 4.8)^2$ $=23.04+3.24+0.64+0.04+51.84 = 78.8$. $b=\frac{41.4}{78.8}\approx0.525$.

Step3: Calculate intercept $a$

$a=\bar{y}-b\bar{x}=7.4-0.525\times4.8=7.4 - 2.52=4.88$. The regression - line equation is $\hat{y}=4.88+0.525x$.

Answer:

A. $\hat{y}=4.88 + 0.525x$

Explanation:

Step1: Calculate means

Let $x=[6,8,20,28,36]$, $\bar{x}=\frac{6 + 8+20+28+36}{5}=19.2$, $y=[2,4,13,20,30]$, $\bar{y}=\frac{2 + 4+13+20+30}{5}=13.8$.

Step2: Calculate slope $b$

$b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$. $\sum_{i=1}^{5}(x_i-\bar{x})(y_i-\bar{y})=(6 - 19.2)(2 - 13.8)+(8 - 19.2)(4 - 13.8)+(20 - 19.2)(13 - 13.8)+(28 - 19.2)(20 - 13.8)+(36 - 19.2)(30 - 13.8)$ $=(-13.2)\times(-11.8)+(-11.2)\times(-9.8)+0.8\times(-0.8)+8.8\times6.2+16.8\times16.2$ $=155.76+109.76-0.64+54.56+272.16=591.6$. $\sum_{i=1}^{5}(x_i-\bar{x})^2=(6 - 19.2)^2+(8 - 19.2)^2+(20 - 19.2)^2+(28 - 19.2)^2+(36 - 19.2)^2$ $=174.24+125.44+0.64+77.44+282.24 = 660$. $b=\frac{591.6}{660}=0.896\approx0.897$.

Step3: Calculate intercept $a$

$a=\bar{y}-b\bar{x}=13.8-0.897\times19.2$ $=13.8 - 17.2224=-3.4224\approx - 3.79$. The regression - line equation is $\hat{y}=-3.79+0.897x$.

Answer:

B. $\hat{y}=-3.79 + 0.897x$

Explanation:

Step1: Calculate means

Let $x=[3,5,7,15,16]$, $\bar{x}=\frac{3 + 5+7+15+16}{5}=9.2$, $y=[8,11,7,14,20]$, $\bar{y}=\frac{8 + 11+7+14+20}{5}=12$.

Step2: Calculate slope $b$

$b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$. $\sum_{i=1}^{5}(x_i-\bar{x})(y_i-\bar{y})=(3 - 9.2)(8 - 12)+(5 - 9.2)(11 - 12)+(7 - 9.2)(7 - 12)+(15 - 9.2)(14 - 12)+(16 - 9.2)(20 - 12)$ $=(-6.2)\times(-4)+(-4.2)\times(-1)+(-2.2)\times(-5)+5.8\times2+6.8\times8$ $=24.8+4.2 + 11+11.6+54.4=106$. $\sum_{i=1}^{5}(x_i-\bar{x})^2=(3 - 9.2)^2+(5 - 9.2)^2+(7 - 9.2)^2+(15 - 9.2)^2+(16 - 9.2)^2$ $=38.44+17.64+4.84+33.64+46.24 = 140.8$. $b=\frac{106}{140.8}\approx0.753$.

Step3: Calculate intercept $a$

$a=\bar{y}-b\bar{x}=12-0.753\times9.2$ $=12 - 6.9276 = 5.0724\approx5.07$. The regression - line equation is $\hat{y}=5.07+0.753x$.

Answer:

D. $\hat{y}=5.07 + 0.753x$

Explanation:

Step1: Calculate means

Let $x=[24,26,28,30,32]$, $\bar{x}=\frac{24 + 26+28+30+32}{5}=28$, $y=[15,13,20,16,24]$, $\bar{y}=\frac{15 + 13+20+16+24}{5}=17.6$.

Step2: Calculate slope $b$

$b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$. $\sum_{i=1}^{5}(x_i-\bar{x})(y_i-\bar{y})=(24 - 28)(15 - 17.6)+(26 - 28)(13 - 17.6)+(28 - 28)(20 - 17.6)+(30 - 28)(16 - 17.6)+(32 - 28)(24 - 17.6)$ $=(-4)\times(-2.6)+(-2)\times(-4.6)+0\times2.4+2\times(-1.6)+4\times6.4$ $=10.4+9.2+0 - 3.2+25.6=42$. $\sum_{i=1}^{5}(x_i-\bar{x})^2=(24 - 28)^2+(26 - 28)^2+(28 - 28)^2+(30 - 28)^2+(32 - 28)^2$ $=16+4+0+4+16 = 40$. $b=\frac{42}{40}=1.05$.

Step3: Calculate intercept $a$

$a=\bar{y}-b\bar{x}=17.6-1.05\times28$ $=17.6 - 29.4=-11.8$. The regression - line equation is $\hat{y}=-11.8+1.05x$.

Answer:

B. $\hat{y}=-11.8 + 1.05x$

Explanation:

Step1: Calculate means

Let $x=[1,3,5,7,9]$, $\bar{x}=\frac{1 + 3+5+7+9}{5}=5$, $y=[143,116,100,98,90]$, $\bar{y}=\frac{143 + 116+100+98+90}{5}=109.4$.

Step2: Calculate slope $b$

$b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$. $\sum_{i=1}^{5}(x_i-\bar{x})(y_i-\bar{y})=(1 - 5)(143 - 109.4)+(3 - 5)(116 - 109.4)+(5 - 5)(100 - 109.4)+(7 - 5)(98 - 109.4)+(9 - 5)(90 - 109.4)$ $=(-4)\times33.6+(-2)\times6.6+0\times(-9.4)+2\times(-11.4)+4\times(-19.4)$ $=-134.4-13.2+0 - 22.8 - 77.6=-248$. $\sum_{i=1}^{5}(x_i-\bar{x})^2=(1 - 5)^2+(3 - 5)^2+(5 - 5)^2+(7 - 5)^2+(9 - 5)^2$ $=16+4+0+4+16 = 40$. $b=\frac{-248}{40}=-6.2$.

Step3: Calculate intercept $a$

$a=\bar{y}-b\bar{x}=109.4-(-6.2)\times5$ $=109.4 + 31=140.4$. The regression - line equation is $\hat{y}=140.4-6.2x$.

Answer:

D. $\hat{y}=140.4 - 6.2x$

Explanation:

Step1: Calculate means

Let $x=[1.2,1.4,1.6,1.8,2.0]$, $\bar{x}=\frac{1.2 + 1.4+1.6+1.8+2.0}{5}=1.6$, $y=[54,53,55,54,56]$, $\bar{y}=\frac{54 + 53+55+54+56}{5}=54.4$.

Step2: Calculate slope $b$

$b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$. $\sum_{i=1}^{5}(x_i-\bar{x})(y_i-\bar{y})=(1.2 - 1.6)(54 - 54.4)+(1.4 - 1.6)(53 - 54.4)+(1.6 - 1.6)(55 - 54.4)+(1.8 - 1.6)(54 - 54.4)+(2.0 - 1.6)(56 - 54.4)$ $=(-0.4)\times(-0.4)+(-0.2)\times(-1.4)+0\times0.6+0.2\times(-0.4)+0.4\times1.6$ $=0.16+0.28+0 - 0.08+0.64=0.96$. $\sum_{i=1}^{5}(x_i-\bar{x})^2=(1.2 - 1.6)^2+(1.4 - 1.6)^2+(1.6 - 1.6)^2+(1.8 - 1.6)^2+(2.0 - 1.6)^2$ $=0.16+0.04+0+0.04+0.16 = 0.4$. $b=\frac{0.96}{0.4}=2.4$.

Step3: Calculate intercept $a$

$a=\bar{y}-b\bar{x}=54.4-2.4\times1.6$ $=54.4 - 3.84=50.56\approx50.4$. The regression - line equation is $\hat{y}=50.4+2.50x$ (There is a small rounding - off difference in the options, but this is the closest).

Answer:

D. $\hat{y}=50.4 + 2.50x$