find the equation of the regression line for the given data. then construct a scatter plot of the data and…

find the equation of the regression line for the given data. then construct a scatter plot of the data and draw the regression line (the pair of variables have a significant correlation.) then use the regression equation to predict the value of y for each of the given x - values, if meaningful. the table below shows the heights (in feet) and the number of stories of six notable buildings in a city.\n| height, x | 772 | 628 | 518 | 508 | 496 | 483 |\n| stories, y | 51 | 48 | 46 | 41 | 38 | 37 |\n(a) x = 501 feet\n(b) x = 641 feet\n(c) x = 315 feet\n(d) x = 726 feet\nfind the regression equation\n\\(\\hat{y}=\\square x+\\square\\)\n(round the slope to three decimal places as needed. round the y - intercept to two decimal places as needed.)

find the equation of the regression line for the given data. then construct a scatter plot of the data and draw the regression line (the pair of variables have a significant correlation.) then use the regression equation to predict the value of y for each of the given x - values, if meaningful. the table below shows the heights (in feet) and the number of stories of six notable buildings in a city.\n| height, x | 772 | 628 | 518 | 508 | 496 | 483 |\n| stories, y | 51 | 48 | 46 | 41 | 38 | 37 |\n(a) x = 501 feet\n(b) x = 641 feet\n(c) x = 315 feet\n(d) x = 726 feet\nfind the regression equation\n\\(\\hat{y}=\\square x+\\square\\)\n(round the slope to three decimal places as needed. round the y - intercept to two decimal places as needed.)

Answer

Explanation:

Step1: Calculate necessary sums

Let (n = 6). Calculate (\sum x=772 + 628+518+508+496+483=3405), (\sum y=51 + 48+46+41+38+37 = 261), (\sum xy=772\times51+628\times48 + 518\times46+508\times41+496\times38+483\times37=772\times51+628\times48+518\times46+508\times41+496\times38+483\times37 = 147793), (\sum x^{2}=772^{2}+628^{2}+518^{2}+508^{2}+496^{2}+483^{2}=1979947).

Step2: Calculate the slope (b_1)

The formula for the slope (b_1=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}). [ \begin{align*} b_1&=\frac{6\times147793 - 3405\times261}{6\times1979947-3405^{2}}\ &=\frac{886758-888705}{11879682 - 11594025}\ &=\frac{-1947}{285657}\approx - 0.007 \end{align*} ]

Step3: Calculate the y - intercept (b_0)

The formula for the y - intercept (b_0=\bar{y}-b_1\bar{x}), where (\bar{x}=\frac{\sum x}{n}=\frac{3405}{6}=567.5) and (\bar{y}=\frac{\sum y}{n}=\frac{261}{6}=43.5). [ \begin{align*} b_0&=43.5-(-0.007)\times567.5\ &=43.5 + 3.9725\ &\approx47.47 \end{align*} ] The regression equation is (\hat{y}=-0.007x + 47.47)

Step4: Predict values of y

(a) When (x = 501)

(\hat{y}=-0.007\times501+47.47=-3.507 + 47.47=43.963\approx44)

(b) When (x = 641)

(\hat{y}=-0.007\times641+47.47=-4.487+47.47 = 42.983\approx43)

(c) When (x = 315)

(\hat{y}=-0.007\times315+47.47=-2.205 + 47.47=45.265\approx45)

(d) When (x = 726)

(\hat{y}=-0.007\times726+47.47=-5.082+47.47 = 42.388\approx42)

Answer:

The regression equation is (\hat{y}=-0.007x + 47.47) (a) 44 (b) 43 (c) 45 (d) 42