Question

writing the equation of the trend line
enter values for x and y in the table to plot points in the graph. use the regression calculator to compare the teams’ number of runs with their number of wins.
r=runs; w=wins
the equation of the trend line is
y = -0.15x + 23.21
y = -0.15x - 23.21
y = 0.15x + 23.21
y = 0.15x - 23.21

Explanation:

Step1: Analyze the relationship between R (x) and W (y)

From the data, as R (runs) increases, W (wins) generally increases? Wait, no, let's check the slope. Let's take two points. For example, when R = 808, W = 93; R = 563, W = 55. So as R decreases, W decreases. So the slope should be positive? Wait, no, wait: let's calculate the slope between (808,93) and (563,55). Slope $m=\frac{55 - 93}{563 - 808}=\frac{- 38}{-245}\approx0.155$, which is positive. So the slope is positive, around 0.15. Now check the y - intercept. Let's use point (808,93) and slope 0.15. $y=0.15x + b$. So $93 = 0.15\times808 + b$. $0.15\times808 = 121.2$. Then $b=93 - 121.2=-28.2$, but the options have 23.21. Wait, maybe another point. Let's take (609,69). $y = 0.15x + b$. $69=0.15\times609 + b$. $0.15\times609 = 91.35$. $b=69 - 91.35=-22.35$. Close to -23.21? Wait, no, maybe I mixed x and y. Wait, the x - axis is R (runs), y - axis is W (wins). Wait, maybe the equation is W (y) = 0.15*R (x)+23.21? Wait, no, when x is 600, y would be 0.15*600 + 23.21=90 + 23.21=113.21, which is too big. Wait, no, maybe the slope is positive, but let's check the options. The options are $y=-0.15x + 23.21$, $y=-0.15x - 23.21$, $y = 0.15x+23.21$, $y = 0.15x - 23.21$. Wait, when x increases, if slope is positive, y increases. Let's check with x = 808: $y = 0.15*808+23.21=121.2 + 23.21 = 144.41$, which is more than 93. No. Wait, maybe I got x and y reversed. Wait, maybe x is W and y is R? No, the problem says "the equation of the trend line" for R (x) and W (y). Wait, maybe the slope is negative? Wait, let's recalculate the slope between (808,93) and (768,94). Wait, 808 to 768 (decrease in x), 93 to 94 (increase in y). So slope is $\frac{94 - 93}{768 - 808}=\frac{1}{-40}=-0.025$, no. Wait, maybe I made a mistake. Wait, let's use regression. Let's calculate the mean of x (R) and y (W). Mean of R: (808 + 768+655+684+637+619+613+609+563)/9. Let's sum R: 808+768=1576; +655=2231; +684=2915; +637=3552; +619=4171; +613=4784; +609=5393; +563=5956. Mean x = 5956/9≈661.78. Mean y (W): (93+94+66+81+86+75+61+69+55)/9. Sum W: 93+94=187; +66=253; +81=334; +86=420; +75=495; +61=556; +69=625; +55=680. Mean y = 680/9≈75.56. Now, the slope $m=\frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2}$. Let's calculate (x_i - \bar{x}) for R: 808 - 661.78≈146.22; 768 - 661.78≈106.22; 655 - 661.78≈-6.78; 684 - 661.78≈22.22; 637 - 661.78≈-24.78; 619 - 661.78≈-42.78; 613 - 661.78≈-48.78; 609 - 661.78≈-52.78; 563 - 661.78≈-98.78. (y_i - \bar{y}) for W: 93 - 75.56≈17.44; 94 - 75.56≈18.44; 66 - 75.56≈-9.56; 81 - 75.56≈5.44; 86 - 75.56≈10.44; 75 - 75.56≈-0.56; 61 - 75.56≈-14.56; 69 - 75.56≈-6.56; 55 - 75.56≈-20.56. Now calculate numerator: (146.22*17.44)+(106.22*18.44)+(-6.78* - 9.56)+(22.22*5.44)+(-24.78*10.44)+(-42.78* - 0.56)+(-48.78* - 14.56)+(-52.78* - 6.56)+(-98.78* - 20.56). Let's compute each term: 146.22*17.44≈2550; 106.22*18.44≈1959; -6.78* - 9.56≈64.8; 22.22*5.44≈120.9; -24.78*10.44≈-258.7; -42.78* - 0.56≈24.0; -48.78* - 14.56≈710.3; -52.78* - 6.56≈346.2; -98.78* - 20.56≈2031. Sum numerator: 2550+1959=4509+64.8=4573.8+120.9=4694.7 - 258.7=4436+24=4460+710.3=5170.3+346.2=5516.5+2031=7547.5. Denominator: sum of (x_i - \bar{x})^2. 146.22²≈21380; 106.22²≈11283; (-6.78)²≈46; 22.22²≈494; (-24.78)²≈614; (-42.78)²≈1820; (-48.78)²≈2380; (-52.78)²≈2786; (-98.78)²≈9758. Sum: 21380+11283=32663+46=32709+494=33203+614=33817+1820=35637+2380=38017+2786=40803+9758=50561. So slope $m=\frac{7547.5}{50561}\approx0.149\approx0.15$. Now, y - intercept $b=\bar{y}-m\bar{x}=75.56 - 0.15\times661.78=75.56 - 99.27=-23.71\approx - 23.21$. Wait, so the equation is $y = 0.15x - 23.21$? Wait, but when x = 661.78 (mean x), y = 75.56. Let's check $y = 0.15x - 23.21$: 0.15*661.78≈99.27 - 23.21≈76.06, close to 75.56 (due to rounding). Now check the sign of the slope. Since as x (R) increases, y (W) increases (because slope is positive), so the equation should be $y = 0.15x - 23.21$. Wait, but let's check with a point. Take x = 808: $y = 0.15*808 - 23.21=121.2 - 23.21=97.99$, close to 93? No, that's not close. Wait, maybe I mixed x and y. Wait, maybe the independent variable is W and dependent is R? No, the problem says "number of runs with their number of wins", so R is x, W is y. Wait, maybe the slope is negative? Wait, no, the calculation of slope from the mean gave positive. Wait, maybe the options have a typo, but among the options, $y = 0.15x - 23.21$ is the one with positive slope and y - intercept close to our calculation. Wait, but when x = 609, $y = 0.15*609 - 23.21=91.35 - 23.21=68.14$, which is close to 69. That's good. When x = 563, $y = 0.15*563 - 23.21=84.45 - 23.21=61.24$, close to 55? No, 55 is lower. Wait, maybe the correct equation is $y = 0.15x - 23.21$.

Step2: Verify with another point

Take x = 619 (R = 619), y should be W = 75. $y = 0.15*619 - 23.21=92.85 - 23.21=69.64$, not 75. Hmm. Wait, maybe the slope is negative. Let's try $y=-0.15x + 23.21$. For x = 808: $y=-0.15*808 + 23.21=-121.2 + 23.21=-97.99$, which is wrong. $y=-0.15x - 23.21$: x = 808, $y=-121.2 - 23.21=-144.41$, wrong. $y = 0.15x+23.21$: x = 808, $y=121.2 + 23.21=144.41$, wrong. Wait, I must have messed up the variables. Wait, maybe the x - axis is W and y - axis is R? Let's check. If x is W (wins), y is R (runs). Then the equation would be R = 0.15W + 23.21? No, that doesn't make sense. Wait, the graph has x - axis from 600 to 800, which is R (runs), and y - axis from 60 to 90, which is W (wins). So when x (R) is 600, y (W) should be around, from the graph, maybe 60 - 90. Let's check the equation $y = 0.15x - 23.21$ when x = 600: $y=90 - 23.21=66.79$, which is in the 60 - 90 range. When x = 800: $y=120 - 23.21=96.79$, which is close to 93,94. So that's better. So the correct equation is $y = 0.15x - 23.21$.

Answer:

$y = 0.15x - 23.21$